From 007a78144217ead600a7527e13d71a0b3380fc31 Mon Sep 17 00:00:00 2001 From: Kamal Saleh Date: Tue, 28 Nov 2023 09:57:28 +0100 Subject: [PATCH] Start using FpCategories infrastructure (quivers, pathcategories and their linear closures or quotients) bump versions HomotopyCategories to V2023.12-01 DerivedCategories to V2023.12-01 --- DerivedCategories/PackageInfo.g | 4 +- .../examples/notebooks/HappelTheorem.ipynb | 851 ++++++++++-------- .../notebooks/TiltingEquivalence.ipynb | 771 ++++++++-------- .../gap/OnlyWithFunctorCategories.gi | 11 +- .../tst/happel-theorem-in-copresheaves.tst | 31 +- HomotopyCategories/PackageInfo.g | 2 +- ...uence_in_homotopy_category_of_k_rows.ipynb | 8 +- HomotopyCategories/gap/OnlyWithAlgebroids.gi | 42 +- HomotopyCategories/tst/TiltingEquivalence.tst | 18 +- 9 files changed, 918 insertions(+), 820 deletions(-) diff --git a/DerivedCategories/PackageInfo.g b/DerivedCategories/PackageInfo.g index 0f44d389..67110dbe 100644 --- a/DerivedCategories/PackageInfo.g +++ b/DerivedCategories/PackageInfo.g @@ -10,7 +10,7 @@ SetPackageInfo( rec( PackageName := "DerivedCategories", Subtitle := "Derived categories of Abelian categories", -Version := "2023.11-02", +Version := "2023.12-01", Date := (function ( ) if IsBound( GAPInfo.SystemEnvironment.GAP_PKG_RELEASE_DATE ) then return GAPInfo.SystemEnvironment.GAP_PKG_RELEASE_DATE; else return Concatenation( ~.Version{[ 1 .. 4 ]}, "-", ~.Version{[ 6, 7 ]}, "-01" ); fi; end)( ), License := "GPL-2.0-or-later", @@ -73,7 +73,7 @@ Dependencies := rec( NeededOtherPackages := [ [ "CAP", ">= 2022.09-17" ], [ "SubcategoriesForCAP", ">= 2020.10-02" ], - [ "HomotopyCategories", ">= 2021.11-02" ], + [ "HomotopyCategories", ">= 2023.12-01" ], [ "ToolsForHigherHomologicalAlgebra", ">= 2020.10-02" ], #[ "PreSheaves", ">= 2022.11-04"], ], diff --git a/DerivedCategories/examples/notebooks/HappelTheorem.ipynb b/DerivedCategories/examples/notebooks/HappelTheorem.ipynb index 4994849c..1ee19dd0 100644 --- a/DerivedCategories/examples/notebooks/HappelTheorem.ipynb +++ b/DerivedCategories/examples/notebooks/HappelTheorem.ipynb @@ -50,7 +50,7 @@ { "data": { "text/plain": [ - "GAP: q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]" + "GAP: FinQuiver( \"q(v1,v2,v3,v4)[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4]\" )" ] }, "execution_count": 3, @@ -59,7 +59,7 @@ } ], "source": [ - "๐“บ = RightQuiver( \"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" )" + "๐“บ = FinQuiver( g\"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" )" ] }, { @@ -69,9 +69,8 @@ "metadata": {}, "outputs": [], "source": [ - "SetLabelsAsLaTeXStrings( ๐“บ, \n", - " ConvertJuliaToGAP( [\"v_{1}\", \"v_{2}\", \"v_{3}\", \"v_{4}\"] ),\n", - " ConvertJuliaToGAP( [\"a\", \"b\", \"c\", \"d\"] ) )" + "SetLaTeXStringsOfObjects( ๐“บ, ConvertJuliaToGAP( [\"v_1\", \"v_2\", \"v_3\", \"v_4\"] ) )\n", + "SetLaTeXStringsOfMorphisms( ๐“บ, ConvertJuliaToGAP( [\"a\", \"b\", \"c\", \"d\"] ) )" ] }, { @@ -83,7 +82,7 @@ { "data": { "text/plain": [ - "GAP: FreeCategory( RightQuiver( \"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) )" + "GAP: PathCategory( FinQuiver( \"q(v1,v2,v3,v4)[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4]\" ) )" ] }, "execution_count": 5, @@ -92,7 +91,7 @@ } ], "source": [ - "F๐“บ = FreeCategory( ๐“บ )" + "F๐“บ = PathCategory( ๐“บ )" ] }, { @@ -125,7 +124,7 @@ { "data": { "text/plain": [ - "GAP: Algebroid( Q, FreeCategory( RightQuiver( \"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) )" + "GAP: Q-LinearClosure( PathCategory( FinQuiver( \"q(v1,v2,v3,v4)[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4]\" ) ) )" ] }, "execution_count": 7, @@ -147,7 +146,7 @@ "data": { "text/plain": [ "1-element Vector{GapObj}:\n", - " GAP: (v1)-[-1*(c*d) + 1*(a*b)]->(v4)" + " GAP: 1*aโ€ขb - 1*cโ€ขd:(v1) -โ‰ป (v4)" ] }, "execution_count": 8, @@ -168,7 +167,7 @@ { "data": { "text/plain": [ - "GAP: Algebroid( Q, FreeCategory( RightQuiver( \"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations" + "GAP: Q-algebroid( {v1,v2,v3,v4}[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4] ) defined by 4 objects and 4 generating morphisms" ] }, "execution_count": 9, @@ -177,7 +176,7 @@ } ], "source": [ - "๐€ = kF๐“บ / ฯ" + "๐€ = AlgebroidFromDataTables( kF๐“บ / ฯ )" ] }, { @@ -190,13 +189,11 @@ "name": "stdout", "output_type": "stream", "text": [ - "A CAP category with name Algebroid( Q, FreeCategory( RightQuiver( \"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations:\n", + "A CAP category with name Q-algebroid( {v1,v2,v3,v4}[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4] ) defined by 4 objects and 4 generating morphisms:\n", "\n", "27 primitive operations were used to derive 80 operations for this category which algorithmically\n", "* IsEquippedWithHomomorphismStructure\n", - "* IsLinearCategoryOverCommutativeRing\n", - "and furthermore mathematically\n", - "* IsFinitelyPresentedLinearCategory\n" + "* IsLinearCategoryOverCommutativeRing\n" ] } ], @@ -207,6 +204,27 @@ { "cell_type": "code", "execution_count": 11, + "id": "7a74481d-fbcd-46e9-a9b7-df22d6048c02", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "true" + ] + }, + "execution_count": 11, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "IsAdmissibleAlgebroid( ๐€ )" + ] + }, + { + "cell_type": "code", + "execution_count": 12, "id": "d664448e", "metadata": {}, "outputs": [ @@ -216,7 +234,7 @@ "GAP: <(v4)>" ] }, - "execution_count": 11, + "execution_count": 12, "metadata": {}, "output_type": "execute_result" } @@ -227,7 +245,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 13, "id": "dc2a34fd", "metadata": {}, "outputs": [ @@ -237,7 +255,7 @@ "GAP: [ <(v1)>, <(v2)>, <(v3)>, <(v4)> ]" ] }, - "execution_count": 12, + "execution_count": 13, "metadata": {}, "output_type": "execute_result" } @@ -248,17 +266,17 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 14, "id": "6a1fdcfd", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: [ (v1)-[{ 1*(a) }]->(v2), (v2)-[{ 1*(b) }]->(v4), (v1)-[{ 1*(c) }]->(v3), (v3)-[{ 1*(d) }]->(v4) ]" + "GAP: [ <1*a:(v1) -โ‰ป (v2)>, <1*b:(v2) -โ‰ป (v4)>, <1*c:(v1) -โ‰ป (v3)>, <1*d:(v3) -โ‰ป (v4)> ]" ] }, - "execution_count": 13, + "execution_count": 14, "metadata": {}, "output_type": "execute_result" } @@ -269,17 +287,17 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 15, "id": "0b241e61", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: [ (v1)-[{ 1*(a*b) }]->(v4) ]" + "GAP: [ <1*cโ€ขd:(v1) -โ‰ป (v4)> ]" ] }, - "execution_count": 14, + "execution_count": 15, "metadata": {}, "output_type": "execute_result" } @@ -290,14 +308,14 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 16, "id": "911a595e", "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "$${v_{1}}-\\left({{ab}}\\right)\\rightarrow{v_{4}}$$" + "$${v_1}-\\left({{c}{d}}\\right)\\rightarrow{v_4}$$" ] }, "metadata": {}, @@ -310,17 +328,59 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 17, + "id": "898f612e-722d-4f79-8737-31e73a18b012", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "GAP: Q-algebroid( {v1,v2,v3,v4}[a:v2-โ‰ปv1,b:v4-โ‰ปv2,c:v3-โ‰ปv1,d:v4-โ‰ปv3] ) defined by 4 objects and 4 generating morphisms" + ] + }, + "execution_count": 17, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "๐€แต’แต– = OppositeAlgebroid( ๐€ )" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "id": "5d936d7c-3c57-4ccd-b064-53fa57e143de", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "true" + ] + }, + "execution_count": 18, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "IsAdmissibleAlgebroid( ๐€แต’แต– )" + ] + }, + { + "cell_type": "code", + "execution_count": 19, "id": "22acb10b", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations, Rows( Q ) )" + "GAP: PreSheaves( Q-algebroid( {v1,v2,v3,v4}[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4] ) defined by 4 objects and 4 generating morphisms, Rows( Q ) )" ] }, - "execution_count": 16, + "execution_count": 19, "metadata": {}, "output_type": "execute_result" } @@ -331,7 +391,31 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 20, + "id": "187e6ce3-58ab-4b29-a242-8a93346a56de", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "A CAP category with name PreSheaves( Q-algebroid( {v1,v2,v3,v4}[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4] ) defined by 4 objects and 4 generating morphisms, Rows( Q ) ):\n", + "\n", + "98 primitive operations were used to derive 347 operations for this category which algorithmically\n", + "* IsEquippedWithHomomorphismStructure\n", + "* IsLinearCategoryOverCommutativeRing\n", + "* IsAbelianCategoryWithEnoughInjectives\n", + "* IsAbelianCategoryWithEnoughProjectives\n" + ] + } + ], + "source": [ + "Display( PSh )" + ] + }, + { + "cell_type": "code", + "execution_count": 21, "id": "f4290d7d", "metadata": {}, "outputs": [ @@ -341,7 +425,7 @@ "GAP: Yoneda embedding functor" ] }, - "execution_count": 17, + "execution_count": 21, "metadata": {}, "output_type": "execute_result" } @@ -352,7 +436,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 22, "id": "3c4d0aca", "metadata": {}, "outputs": [ @@ -362,10 +446,10 @@ "text": [ "Yoneda embedding functor:\n", "\n", - "Algebroid( Q, FreeCategory( RightQuiver( \"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations\n", + "Q-algebroid( {v1,v2,v3,v4}[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4] ) defined by 4 objects and 4 generating morphisms\n", " |\n", " V\n", - "PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations, Rows( Q ) )\n" + "PreSheaves( Q-algebroid( {v1,v2,v3,v4}[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4] ) defined by 4 objects and 4 generating morphisms, Rows( Q ) )\n" ] } ], @@ -375,7 +459,7 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 23, "id": "82cd8f2d", "metadata": {}, "outputs": [ @@ -385,7 +469,7 @@ "GAP: <(v1)->1, (v2)->0, (v3)->0, (v4)->0; (a)->0x1, (b)->0x0, (c)->0x1, (d)->0x0>" ] }, - "execution_count": 19, + "execution_count": 23, "metadata": {}, "output_type": "execute_result" } @@ -396,7 +480,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 24, "id": "0172c3ec", "metadata": {}, "outputs": [ @@ -404,7 +488,7 @@ "data": { "text/latex": [ "$$\\begin{array}{ccc}\n", - " v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{b} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\{c} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{d} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\\\end{array}$$" + " v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{b} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\{c} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{d} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\\\end{array}$$" ] }, "metadata": {}, @@ -417,7 +501,7 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 25, "id": "cbf1b6d4", "metadata": {}, "outputs": [ @@ -427,7 +511,7 @@ "GAP: <(v1)->1, (v2)->1, (v3)->0, (v4)->0; (a)->1x1, (b)->0x1, (c)->0x1, (d)->0x0>" ] }, - "execution_count": 21, + "execution_count": 25, "metadata": {}, "output_type": "execute_result" } @@ -438,7 +522,7 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 26, "id": "f1366069", "metadata": {}, "outputs": [ @@ -446,7 +530,7 @@ "data": { "text/latex": [ "$$\\begin{array}{ccc}\n", - " v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + " v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " 1 \n", "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{b} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{c} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{d} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\\\end{array}$$" ] @@ -461,7 +545,7 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 27, "id": "74b72bf1", "metadata": {}, "outputs": [ @@ -471,7 +555,7 @@ "GAP: <(v1)->1, (v2)->0, (v3)->1, (v4)->0; (a)->0x1, (b)->0x0, (c)->1x1, (d)->0x1>" ] }, - "execution_count": 23, + "execution_count": 27, "metadata": {}, "output_type": "execute_result" } @@ -482,7 +566,7 @@ }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 28, "id": "0dc43dd6", "metadata": {}, "outputs": [ @@ -490,7 +574,7 @@ "data": { "text/latex": [ "$$\\begin{array}{ccc}\n", - " v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{b} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\{c} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + " v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{b} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\{c} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " 1 \n", "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{d} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\\\end{array}$$" ] @@ -505,7 +589,7 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 29, "id": "3464b123", "metadata": {}, "outputs": [ @@ -515,7 +599,7 @@ "GAP: <(v1)->1, (v2)->1, (v3)->1, (v4)->1; (a)->1x1, (b)->1x1, (c)->1x1, (d)->1x1>" ] }, - "execution_count": 25, + "execution_count": 29, "metadata": {}, "output_type": "execute_result" } @@ -526,7 +610,7 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 30, "id": "79398022", "metadata": {}, "outputs": [ @@ -534,7 +618,7 @@ "data": { "text/latex": [ "$$\\begin{array}{ccc}\n", - " v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + " v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " 1 \n", "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{b} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " 1 \n", @@ -555,7 +639,7 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 31, "id": "70a85370", "metadata": {}, "outputs": [ @@ -565,7 +649,7 @@ "true" ] }, - "execution_count": 27, + "execution_count": 31, "metadata": {}, "output_type": "execute_result" } @@ -576,17 +660,17 @@ }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 32, "id": "681c7774", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations )" + "GAP: AdditiveClosure( Q-algebroid( {v1,v2,v3,v4}[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4] ) defined by 4 objects and 4 generating morphisms )" ] }, - "execution_count": 28, + "execution_count": 32, "metadata": {}, "output_type": "execute_result" } @@ -597,7 +681,7 @@ }, { "cell_type": "code", - "execution_count": 29, + "execution_count": 33, "id": "aaceabc2", "metadata": {}, "outputs": [ @@ -607,7 +691,7 @@ "GAP: Extension of Yoneda embedding functor to a functor from the additive closure of the source" ] }, - "execution_count": 29, + "execution_count": 33, "metadata": {}, "output_type": "execute_result" } @@ -618,7 +702,7 @@ }, { "cell_type": "code", - "execution_count": 30, + "execution_count": 34, "id": "cfad7ed4", "metadata": {}, "outputs": [ @@ -628,10 +712,10 @@ "text": [ "Extension of Yoneda embedding functor to a functor from the additive closure of the source:\n", "\n", - "AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations )\n", + "AdditiveClosure( Q-algebroid( {v1,v2,v3,v4}[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4] ) defined by 4 objects and 4 generating morphisms )\n", " |\n", " V\n", - "PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations, Rows( Q ) )\n" + "PreSheaves( Q-algebroid( {v1,v2,v3,v4}[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4] ) defined by 4 objects and 4 generating morphisms, Rows( Q ) )\n" ] } ], @@ -641,17 +725,17 @@ }, { "cell_type": "code", - "execution_count": 31, + "execution_count": 35, "id": "2a0382b3", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: Homotopy category by cochains( AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations ) )" + "GAP: Homotopy category by cochains( AdditiveClosure( Q-algebroid( {v1,v2,v3,v4}[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4] ) defined by 4 objects and 4 generating morphisms ) )" ] }, - "execution_count": 31, + "execution_count": 35, "metadata": {}, "output_type": "execute_result" } @@ -662,7 +746,7 @@ }, { "cell_type": "code", - "execution_count": 32, + "execution_count": 36, "id": "4007703a", "metadata": {}, "outputs": [ @@ -672,7 +756,7 @@ "GAP: Extension of ( Extension of Yoneda embedding functor to a functor from the additive closure of the source ) to homotopy categories by cochains" ] }, - "execution_count": 32, + "execution_count": 36, "metadata": {}, "output_type": "execute_result" } @@ -683,7 +767,7 @@ }, { "cell_type": "code", - "execution_count": 33, + "execution_count": 37, "id": "41dd9ad7", "metadata": {}, "outputs": [ @@ -693,10 +777,10 @@ "text": [ "Extension of ( Extension of Yoneda embedding functor to a functor from the additive closure of the source ) to homotopy categories by cochains:\n", "\n", - "Homotopy category by cochains( AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations ) )\n", + "Homotopy category by cochains( AdditiveClosure( Q-algebroid( {v1,v2,v3,v4}[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4] ) defined by 4 objects and 4 generating morphisms ) )\n", " |\n", " V\n", - "Homotopy category by cochains( PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations, Rows( Q ) ) )\n" + "Homotopy category by cochains( PreSheaves( Q-algebroid( {v1,v2,v3,v4}[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4] ) defined by 4 objects and 4 generating morphisms, Rows( Q ) ) )\n" ] } ], @@ -706,17 +790,17 @@ }, { "cell_type": "code", - "execution_count": 34, + "execution_count": 38, "id": "4aa913e4", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations ) defined by 1 underlying objects>" + "GAP: " ] }, - "execution_count": 34, + "execution_count": 38, "metadata": {}, "output_type": "execute_result" } @@ -727,17 +811,17 @@ }, { "cell_type": "code", - "execution_count": 35, + "execution_count": 39, "id": "214f5d0c", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations ) defined by 2 underlying objects>" + "GAP: " ] }, - "execution_count": 35, + "execution_count": 39, "metadata": {}, "output_type": "execute_result" } @@ -748,17 +832,17 @@ }, { "cell_type": "code", - "execution_count": 36, + "execution_count": 40, "id": "71bb6625", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations ) defined by a 1 x 2 matrix of underlying morphisms>" + "GAP: " ] }, - "execution_count": 36, + "execution_count": 40, "metadata": {}, "output_type": "execute_result" } @@ -772,14 +856,14 @@ }, { "cell_type": "code", - "execution_count": 37, + "execution_count": 41, "id": "6e3eaedc", "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "$${v_{1}}\\xrightarrow{\\begin{pmatrix}{a}&{c}\\end{pmatrix}}{v_{2}}\\oplus{v_{3}}$$" + "$${v_1}\\xrightarrow{\\begin{pmatrix}{a}&{c}\\end{pmatrix}}{v_2}\\oplus{v_3}$$" ] }, "metadata": {}, @@ -792,17 +876,17 @@ }, { "cell_type": "code", - "execution_count": 38, + "execution_count": 42, "id": "a8d90125", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations ) ) supported on the interval [ -1 .. 0 ]>" + "GAP: " ] }, - "execution_count": 38, + "execution_count": 42, "metadata": {}, "output_type": "execute_result" } @@ -813,7 +897,7 @@ }, { "cell_type": "code", - "execution_count": 39, + "execution_count": 43, "id": "932b5284", "metadata": {}, "outputs": [ @@ -821,7 +905,7 @@ "data": { "text/latex": [ "$$\\begin{array}{c}\n", - "{v_{2}}\\oplus{v_{3}}\n", + "{v_2}\\oplus{v_3}\n", "\\\\\n", "\\uparrow_{\\phantom{-1}}\n", "\\\\\n", @@ -830,7 +914,7 @@ "{\\vert_{-1}}\n", "\n", "\\\\\n", - "{v_{1}}\\end{array}$$" + "{v_1}\\end{array}$$" ] }, "metadata": {}, @@ -843,17 +927,17 @@ }, { "cell_type": "code", - "execution_count": 40, + "execution_count": 44, "id": "d8835d01", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations, Rows( Q ) ) ) supported on the interval [ -1 .. 0 ]>" + "GAP: " ] }, - "execution_count": 40, + "execution_count": 44, "metadata": {}, "output_type": "execute_result" } @@ -864,7 +948,7 @@ }, { "cell_type": "code", - "execution_count": 41, + "execution_count": 45, "id": "e255f0e4", "metadata": {}, "outputs": [ @@ -873,7 +957,7 @@ "text/latex": [ "$$\\begin{array}{c}\n", "\\begin{array}{ccc}\n", - " v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 2} \\\\ v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", + " v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 2} \\\\ v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", " 1 & \\cdot \n", "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 2} \\\\ & & \\\\{b} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{c} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", " \\cdot & 1 \n", @@ -882,15 +966,15 @@ "\\uparrow_{\\phantom{-1}}\n", "\\\\\n", "\\begin{array}{ccc}\n", - "v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", + "v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", " 1 & 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 2} \\\\ & & \\\\v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\\\end{array}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 2} \\\\ & & \\\\v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\\\end{array}\n", "\\\\\n", "{\\vert_{-1}}\n", "\n", "\\\\\n", "\\begin{array}{ccc}\n", - " v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{b} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\{c} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{d} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\\\end{array}\\end{array}$$" + " v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{b} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\{c} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{d} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\\\end{array}\\end{array}$$" ] }, "metadata": {}, @@ -903,7 +987,7 @@ }, { "cell_type": "code", - "execution_count": 42, + "execution_count": 46, "id": "ecd866dc", "metadata": {}, "outputs": [ @@ -913,7 +997,7 @@ "GAP: [ 0 ]" ] }, - "execution_count": 42, + "execution_count": 46, "metadata": {}, "output_type": "execute_result" } @@ -924,7 +1008,7 @@ }, { "cell_type": "code", - "execution_count": 43, + "execution_count": 47, "id": "0d0a4b26", "metadata": {}, "outputs": [ @@ -934,7 +1018,7 @@ "GAP: <(v1)->1, (v2)->1, (v3)->1, (v4)->0; (a)->1x1, (b)->0x1, (c)->1x1, (d)->0x1>" ] }, - "execution_count": 43, + "execution_count": 47, "metadata": {}, "output_type": "execute_result" } @@ -945,7 +1029,7 @@ }, { "cell_type": "code", - "execution_count": 44, + "execution_count": 48, "id": "16e0dc96", "metadata": {}, "outputs": [], @@ -955,7 +1039,7 @@ }, { "cell_type": "code", - "execution_count": 45, + "execution_count": 49, "id": "a8a86b55", "metadata": {}, "outputs": [ @@ -965,7 +1049,7 @@ "GAP: <(v1)->2x1, (v2)->1x1, (v3)->1x1, (v4)->0x0>" ] }, - "execution_count": 45, + "execution_count": 49, "metadata": {}, "output_type": "execute_result" } @@ -976,7 +1060,7 @@ }, { "cell_type": "code", - "execution_count": 46, + "execution_count": 50, "id": "68956d6b", "metadata": {}, "outputs": [ @@ -984,9 +1068,9 @@ "data": { "text/latex": [ "$$\\begin{array}{ccc}\n", - "v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", + "v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", " 1 & 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 2} \\\\ & & \\\\v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\\\end{array}$$" + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 2} \\\\ & & \\\\v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\\\end{array}$$" ] }, "metadata": {}, @@ -999,7 +1083,7 @@ }, { "cell_type": "code", - "execution_count": 47, + "execution_count": 51, "id": "32b96ff4", "metadata": {}, "outputs": [ @@ -1007,14 +1091,14 @@ "data": { "text/latex": [ "$$\\begin{array}{ccc}\n", - "v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + "v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " -1 \\\\ \n", " 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\\\end{array}$$" + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\\\end{array}$$" ] }, "metadata": {}, @@ -1027,17 +1111,17 @@ }, { "cell_type": "code", - "execution_count": 48, + "execution_count": 52, "id": "a2d89dd5", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: Homotopy category by cochains( PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations, Rows( Q ) ) )" + "GAP: Homotopy category by cochains( PreSheaves( Q-algebroid( {v1,v2,v3,v4}[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4] ) defined by 4 objects and 4 generating morphisms, Rows( Q ) ) )" ] }, - "execution_count": 48, + "execution_count": 52, "metadata": {}, "output_type": "execute_result" } @@ -1048,17 +1132,17 @@ }, { "cell_type": "code", - "execution_count": 49, + "execution_count": 53, "id": "dac23f5d", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations, Rows( Q ) ) ) supported on the interval [ -1 .. 1 ]>" + "GAP: " ] }, - "execution_count": 49, + "execution_count": 53, "metadata": {}, "output_type": "execute_result" } @@ -1069,7 +1153,7 @@ }, { "cell_type": "code", - "execution_count": 50, + "execution_count": 54, "id": "3eb5afce", "metadata": {}, "outputs": [ @@ -1079,7 +1163,7 @@ "GAP: [ ]" ] }, - "execution_count": 50, + "execution_count": 54, "metadata": {}, "output_type": "execute_result" } @@ -1090,7 +1174,7 @@ }, { "cell_type": "code", - "execution_count": 51, + "execution_count": 55, "id": "53344c29", "metadata": {}, "outputs": [ @@ -1100,7 +1184,7 @@ "GAP: <(v1)->1, (v2)->1, (v3)->1, (v4)->1; (a)->1x1, (b)->1x1, (c)->1x1, (d)->1x1>" ] }, - "execution_count": 51, + "execution_count": 55, "metadata": {}, "output_type": "execute_result" } @@ -1111,7 +1195,7 @@ }, { "cell_type": "code", - "execution_count": 52, + "execution_count": 56, "id": "84cfea05", "metadata": {}, "outputs": [ @@ -1121,7 +1205,7 @@ "GAP: <(v1)->4, (v2)->3, (v3)->3, (v4)->1; (a)->3x4, (b)->1x3, (c)->3x4, (d)->1x3>" ] }, - "execution_count": 52, + "execution_count": 56, "metadata": {}, "output_type": "execute_result" } @@ -1132,7 +1216,7 @@ }, { "cell_type": "code", - "execution_count": 53, + "execution_count": 57, "id": "9c3d0132", "metadata": {}, "outputs": [ @@ -1140,7 +1224,7 @@ "data": { "text/latex": [ "$$\\begin{array}{ccc}\n", - " v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 4} \\\\ v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 3} \\\\ v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 3} \\\\ v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrrr}\n", + " v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 4} \\\\ v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 3} \\\\ v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 3} \\\\ v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrrr}\n", " 1 & \\cdot & \\cdot & \\cdot \\\\ \n", " \\cdot & \\cdot & -1 & \\cdot \\\\ \n", " \\cdot & \\cdot & \\cdot & 1 \n", @@ -1165,7 +1249,7 @@ }, { "cell_type": "code", - "execution_count": 54, + "execution_count": 58, "id": "1e2ec997", "metadata": {}, "outputs": [ @@ -1175,7 +1259,7 @@ "GAP: " ] }, - "execution_count": 54, + "execution_count": 58, "metadata": {}, "output_type": "execute_result" } @@ -1186,17 +1270,17 @@ }, { "cell_type": "code", - "execution_count": 55, + "execution_count": 59, "id": "54f1e7f0", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: A strong exceptional sequence in PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations, Rows( Q ) )" + "GAP: A strong exceptional sequence in PreSheaves( Q-algebroid( {v1,v2,v3,v4}[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4] ) defined by 4 objects and 4 generating morphisms, Rows( Q ) )" ] }, - "execution_count": 55, + "execution_count": 59, "metadata": {}, "output_type": "execute_result" } @@ -1207,17 +1291,17 @@ }, { "cell_type": "code", - "execution_count": 56, + "execution_count": 60, "id": "03053af2", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) )" + "GAP: Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms" ] }, - "execution_count": 56, + "execution_count": 60, "metadata": {}, "output_type": "execute_result" } @@ -1228,17 +1312,17 @@ }, { "cell_type": "code", - "execution_count": 57, + "execution_count": 61, "id": "89a24af1", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]" + "GAP: FinQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4]\" )" ] }, - "execution_count": 57, + "execution_count": 61, "metadata": {}, "output_type": "execute_result" } @@ -1249,7 +1333,7 @@ }, { "cell_type": "code", - "execution_count": 58, + "execution_count": 62, "id": "aaf6f330", "metadata": {}, "outputs": [ @@ -1259,18 +1343,18 @@ "GAP: [ ]" ] }, - "execution_count": 58, + "execution_count": 62, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "RelationsOfAlgebroid( ๐€_๐“” )" + "DefiningRelations( DefiningCategory( ๐€_๐“” ) )" ] }, { "cell_type": "code", - "execution_count": 59, + "execution_count": 63, "id": "240158a5", "metadata": {}, "outputs": [ @@ -1280,7 +1364,7 @@ "GAP: Isomorphism: strong exceptional sequence โŸถ abstraction algebroid" ] }, - "execution_count": 59, + "execution_count": 63, "metadata": {}, "output_type": "execute_result" } @@ -1291,7 +1375,7 @@ }, { "cell_type": "code", - "execution_count": 60, + "execution_count": 64, "id": "50bb0140", "metadata": {}, "outputs": [ @@ -1301,10 +1385,10 @@ "text": [ "Isomorphism: strong exceptional sequence โŸถ abstraction algebroid:\n", "\n", - "A strong exceptional sequence in PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations, Rows( Q ) )\n", + "A strong exceptional sequence in PreSheaves( Q-algebroid( {v1,v2,v3,v4}[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4] ) defined by 4 objects and 4 generating morphisms, Rows( Q ) )\n", " |\n", " V\n", - "Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) )\n" + "Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms\n" ] } ], @@ -1314,7 +1398,7 @@ }, { "cell_type": "code", - "execution_count": 61, + "execution_count": 65, "id": "c1be4152", "metadata": {}, "outputs": [ @@ -1324,7 +1408,7 @@ "GAP: Isomorphism: abstraction algebroid โŸถ strong exceptional sequence" ] }, - "execution_count": 61, + "execution_count": 65, "metadata": {}, "output_type": "execute_result" } @@ -1335,7 +1419,7 @@ }, { "cell_type": "code", - "execution_count": 62, + "execution_count": 66, "id": "a4928aea", "metadata": {}, "outputs": [ @@ -1345,10 +1429,10 @@ "text": [ "Isomorphism: abstraction algebroid โŸถ strong exceptional sequence:\n", "\n", - "Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) )\n", + "Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms\n", " |\n", " V\n", - "A strong exceptional sequence in PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations, Rows( Q ) )\n" + "A strong exceptional sequence in PreSheaves( Q-algebroid( {v1,v2,v3,v4}[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4] ) defined by 4 objects and 4 generating morphisms, Rows( Q ) )\n" ] } ], @@ -1358,7 +1442,7 @@ }, { "cell_type": "code", - "execution_count": 63, + "execution_count": 67, "id": "62ffc534", "metadata": {}, "outputs": [ @@ -1368,7 +1452,7 @@ "GAP: <(E4)>" ] }, - "execution_count": 63, + "execution_count": 67, "metadata": {}, "output_type": "execute_result" } @@ -1379,7 +1463,7 @@ }, { "cell_type": "code", - "execution_count": 64, + "execution_count": 68, "id": "55d00765", "metadata": {}, "outputs": [ @@ -1389,7 +1473,7 @@ "true" ] }, - "execution_count": 64, + "execution_count": 68, "metadata": {}, "output_type": "execute_result" } @@ -1400,17 +1484,17 @@ }, { "cell_type": "code", - "execution_count": 65, + "execution_count": 69, "id": "c15ebae2", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ), Rows( Q ) )" + "GAP: PreSheaves( Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms, Rows( Q ) )" ] }, - "execution_count": 65, + "execution_count": 69, "metadata": {}, "output_type": "execute_result" } @@ -1421,7 +1505,7 @@ }, { "cell_type": "code", - "execution_count": 66, + "execution_count": 70, "id": "e3678525", "metadata": {}, "outputs": [ @@ -1431,7 +1515,7 @@ "GAP: Hom(T,-) functor" ] }, - "execution_count": 66, + "execution_count": 70, "metadata": {}, "output_type": "execute_result" } @@ -1442,7 +1526,7 @@ }, { "cell_type": "code", - "execution_count": 67, + "execution_count": 71, "id": "b2ce09aa", "metadata": {}, "outputs": [ @@ -1452,10 +1536,10 @@ "text": [ "Hom(T,-) functor:\n", "\n", - "PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(v1,v2,v3,v4)[a:v1->v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations, Rows( Q ) )\n", + "PreSheaves( Q-algebroid( {v1,v2,v3,v4}[a:v1-โ‰ปv2,b:v2-โ‰ปv4,c:v1-โ‰ปv3,d:v3-โ‰ปv4] ) defined by 4 objects and 4 generating morphisms, Rows( Q ) )\n", " |\n", " V\n", - "PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ), Rows( Q ) )\n" + "PreSheaves( Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms, Rows( Q ) )\n" ] } ], @@ -1465,7 +1549,7 @@ }, { "cell_type": "code", - "execution_count": 68, + "execution_count": 72, "id": "d47d77c0", "metadata": {}, "outputs": [ @@ -1475,7 +1559,7 @@ "GAP: -โŠ—T functor" ] }, - "execution_count": 68, + "execution_count": 72, "metadata": {}, "output_type": "execute_result" } @@ -1486,7 +1570,7 @@ }, { "cell_type": "code", - "execution_count": 69, + "execution_count": 73, "id": "528f6ed8", "metadata": {}, "outputs": [ @@ -1496,7 +1580,7 @@ "GAP: Hom(T,-) โŠ— T => Id" ] }, - "execution_count": 69, + "execution_count": 73, "metadata": {}, "output_type": "execute_result" } @@ -1507,7 +1591,7 @@ }, { "cell_type": "code", - "execution_count": 70, + "execution_count": 74, "id": "d3b4a581", "metadata": {}, "outputs": [ @@ -1517,7 +1601,7 @@ "GAP: Id => Hom(T, -โŠ—T)" ] }, - "execution_count": 70, + "execution_count": 74, "metadata": {}, "output_type": "execute_result" } @@ -1528,7 +1612,28 @@ }, { "cell_type": "code", - "execution_count": 71, + "execution_count": 75, + "id": "6c12e12d-d87f-45b1-8df0-fa531b6a08c4", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "GAP: Rows( Q )" + ] + }, + "execution_count": 75, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "target_PSh = Target( PSh )" + ] + }, + { + "cell_type": "code", + "execution_count": 76, "id": "b712a0ff", "metadata": {}, "outputs": [ @@ -1538,24 +1643,24 @@ "GAP: <(v1)->2, (v2)->2, (v3)->1, (v4)->4; (a)->2x2, (b)->4x2, (c)->1x2, (d)->4x1>" ] }, - "execution_count": 71, + "execution_count": 76, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "F = CreatePreSheaf( PSh,\n", - " ConvertJuliaToGAP( [ 2, 2, 1, 4 ] ),\n", + "F = CreatePreSheafByValues( PSh,\n", + " ConvertJuliaToGAP( [ 2 / target_PSh, 2 / target_PSh, 1 / target_PSh, 4 / target_PSh ] ),\n", " ConvertJuliaToGAP(\n", - " [ HomalgMatrix( \"[ [ 0, 1 ], [ 0, 0 ] ]\", 2, 2, k ),\n", - " HomalgMatrix( \"[ [ 0, 0 ], [ 1, 0 ], [ 0, 1 ], [ 0, 0 ] ]\", 4, 2, k ),\n", - " HomalgMatrix( \"[ [ 0, 1 ] ]\", 1, 2, k ),\n", - " HomalgMatrix( \"[ [ 0 ], [ 1 ], [ 0 ], [ 0 ] ]\", 4, 1, k ) ] ) )" + " [ HomalgMatrix( \"[ [ 0, 1 ], [ 0, 0 ] ]\", 2, 2, k ) / target_PSh,\n", + " HomalgMatrix( \"[ [ 0, 0 ], [ 1, 0 ], [ 0, 1 ], [ 0, 0 ] ]\", 4, 2, k ) / target_PSh,\n", + " HomalgMatrix( \"[ [ 0, 1 ] ]\", 1, 2, k ) / target_PSh,\n", + " HomalgMatrix( \"[ [ 0 ], [ 1 ], [ 0 ], [ 0 ] ]\", 4, 1, k ) / target_PSh ] ) )" ] }, { "cell_type": "code", - "execution_count": 72, + "execution_count": 77, "id": "308af7aa", "metadata": {}, "outputs": [ @@ -1565,7 +1670,7 @@ "true" ] }, - "execution_count": 72, + "execution_count": 77, "metadata": {}, "output_type": "execute_result" } @@ -1576,7 +1681,7 @@ }, { "cell_type": "code", - "execution_count": 73, + "execution_count": 78, "id": "455d4ed7", "metadata": {}, "outputs": [ @@ -1584,7 +1689,7 @@ "data": { "text/latex": [ "$$\\begin{array}{ccc}\n", - " v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 2} \\\\ v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 2} \\\\ v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 4} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", + " v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 2} \\\\ v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 2} \\\\ v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 4} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", " \\cdot & 1 \\\\ \n", " \\cdot & \\cdot \n", "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 2} \\\\ & & \\\\{b} & \\mapsto & \\mathbb{Q}^{1 \\times 4}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", @@ -1612,7 +1717,7 @@ }, { "cell_type": "code", - "execution_count": 74, + "execution_count": 79, "id": "da8a0433", "metadata": {}, "outputs": [ @@ -1622,7 +1727,7 @@ "GAP: <(v1)->1, (v2)->2, (v3)->1, (v4)->4; (a)->2x1, (b)->4x2, (c)->1x1, (d)->4x1>" ] }, - "execution_count": 74, + "execution_count": 79, "metadata": {}, "output_type": "execute_result" } @@ -1633,7 +1738,7 @@ }, { "cell_type": "code", - "execution_count": 75, + "execution_count": 80, "id": "d71168d1", "metadata": {}, "outputs": [ @@ -1641,7 +1746,7 @@ "data": { "text/latex": [ "$$\\begin{array}{ccc}\n", - " v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 2} \\\\ v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 4} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + " v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 2} \\\\ v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 4} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " 1 \\\\ \n", " \\cdot \n", "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{b} & \\mapsto & \\mathbb{Q}^{1 \\times 4}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", @@ -1669,7 +1774,7 @@ }, { "cell_type": "code", - "execution_count": 76, + "execution_count": 81, "id": "f2b4515d", "metadata": {}, "outputs": [ @@ -1679,7 +1784,7 @@ "GAP: <(v1)->1x2, (v2)->2x2, (v3)->1x1, (v4)->4x4>" ] }, - "execution_count": 76, + "execution_count": 81, "metadata": {}, "output_type": "execute_result" } @@ -1690,7 +1795,7 @@ }, { "cell_type": "code", - "execution_count": 77, + "execution_count": 82, "id": "83875a24", "metadata": {}, "outputs": [ @@ -1698,14 +1803,14 @@ "data": { "text/latex": [ "$$\\begin{array}{ccc}\n", - "v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", + "v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", " \\cdot & 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 2} \\\\ & & \\\\v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 2} \\\\ & & \\\\v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", " 1 & \\cdot \\\\ \n", " \\cdot & 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 2} \\\\ & & \\\\v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 2} \\\\ & & \\\\v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 4}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrrr}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 4}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrrr}\n", " 1 & \\cdot & \\cdot & \\cdot \\\\ \n", " \\cdot & 1 & \\cdot & \\cdot \\\\ \n", " \\cdot & \\cdot & 1 & \\cdot \\\\ \n", @@ -1723,7 +1828,7 @@ }, { "cell_type": "code", - "execution_count": 78, + "execution_count": 83, "id": "b4e8f187", "metadata": {}, "outputs": [ @@ -1733,24 +1838,24 @@ "GAP: <(v1)->0, (v2)->4, (v3)->2, (v4)->1; (a)->4x0, (b)->1x4, (c)->2x0, (d)->1x2>" ] }, - "execution_count": 78, + "execution_count": 83, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "G = CreatePreSheaf( PSh,\n", - " ConvertJuliaToGAP( [ 0, 4, 2, 1 ] ),\n", + "G = CreatePreSheafByValues( PSh,\n", + " ConvertJuliaToGAP( [ 0 / target_PSh, 4 / target_PSh, 2 / target_PSh, 1 / target_PSh ] ),\n", " ConvertJuliaToGAP(\n", - " [ HomalgZeroMatrix( 4, 0, k ),\n", - " HomalgMatrix( \"[ [ 0, 1, 0, 0 ] ]\", 1, 4, k ),\n", - " HomalgZeroMatrix( 2, 0, k ),\n", - " HomalgMatrix( \"[ [ 1, 0 ] ]\", 1, 2, k ) ] ) )" + " [ HomalgZeroMatrix( 4, 0, k ) / target_PSh,\n", + " HomalgMatrix( \"[ [ 0, 1, 0, 0 ] ]\", 1, 4, k ) / target_PSh,\n", + " HomalgZeroMatrix( 2, 0, k ) / target_PSh,\n", + " HomalgMatrix( \"[ [ 1, 0 ] ]\", 1, 2, k ) / target_PSh ] ) )" ] }, { "cell_type": "code", - "execution_count": 79, + "execution_count": 84, "id": "3e3e2d6a", "metadata": {}, "outputs": [ @@ -1760,7 +1865,7 @@ "GAP: <(v1)->2x0, (v2)->2x4, (v3)->1x2, (v4)->4x1>" ] }, - "execution_count": 79, + "execution_count": 84, "metadata": {}, "output_type": "execute_result" } @@ -1778,7 +1883,7 @@ }, { "cell_type": "code", - "execution_count": 80, + "execution_count": 85, "id": "6b24495a", "metadata": {}, "outputs": [ @@ -1788,7 +1893,7 @@ "true" ] }, - "execution_count": 80, + "execution_count": 85, "metadata": {}, "output_type": "execute_result" } @@ -1799,7 +1904,7 @@ }, { "cell_type": "code", - "execution_count": 81, + "execution_count": 86, "id": "a37fa5fe", "metadata": {}, "outputs": [ @@ -1809,7 +1914,7 @@ "true" ] }, - "execution_count": 81, + "execution_count": 86, "metadata": {}, "output_type": "execute_result" } @@ -1820,7 +1925,7 @@ }, { "cell_type": "code", - "execution_count": 82, + "execution_count": 87, "id": "7311d052", "metadata": {}, "outputs": [ @@ -1830,7 +1935,7 @@ "GAP: Extension of ( Hom(T,-) functor ) to homotopy categories by cochains" ] }, - "execution_count": 82, + "execution_count": 87, "metadata": {}, "output_type": "execute_result" } @@ -1841,7 +1946,7 @@ }, { "cell_type": "code", - "execution_count": 83, + "execution_count": 88, "id": "74f034be", "metadata": {}, "outputs": [ @@ -1851,7 +1956,7 @@ "GAP: Extension of ( -โŠ—T functor ) to homotopy categories by cochains" ] }, - "execution_count": 83, + "execution_count": 88, "metadata": {}, "output_type": "execute_result" } @@ -1862,17 +1967,17 @@ }, { "cell_type": "code", - "execution_count": 84, + "execution_count": 89, "id": "9d78e923", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: Homotopy category by cochains( PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ), Rows( Q ) ) )" + "GAP: Homotopy category by cochains( PreSheaves( Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms, Rows( Q ) ) )" ] }, - "execution_count": 84, + "execution_count": 89, "metadata": {}, "output_type": "execute_result" } @@ -1883,7 +1988,7 @@ }, { "cell_type": "code", - "execution_count": 85, + "execution_count": 90, "id": "2397db0e", "metadata": {}, "outputs": [ @@ -1893,7 +1998,7 @@ "GAP: Extention of (Hom(T,-) โŠ— T => Id) to homotopy categories by cochains" ] }, - "execution_count": 85, + "execution_count": 90, "metadata": {}, "output_type": "execute_result" } @@ -1904,17 +2009,17 @@ }, { "cell_type": "code", - "execution_count": 86, + "execution_count": 91, "id": "489390c2", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations, Rows( Q ) ) ) supported on the interval [ 0 ]>" + "GAP: " ] }, - "execution_count": 86, + "execution_count": 91, "metadata": {}, "output_type": "execute_result" } @@ -1925,17 +2030,17 @@ }, { "cell_type": "code", - "execution_count": 87, + "execution_count": 92, "id": "53e3164c", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations, Rows( Q ) ) ) supported on the interval [ 0 .. 2 ]>" + "GAP: " ] }, - "execution_count": 87, + "execution_count": 92, "metadata": {}, "output_type": "execute_result" } @@ -1946,17 +2051,17 @@ }, { "cell_type": "code", - "execution_count": 88, + "execution_count": 93, "id": "1b718e8c", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations, Rows( Q ) ) ) supported on the interval [ 0 .. 2 ]>" + "GAP: " ] }, - "execution_count": 88, + "execution_count": 93, "metadata": {}, "output_type": "execute_result" } @@ -1967,7 +2072,7 @@ }, { "cell_type": "code", - "execution_count": 89, + "execution_count": 94, "id": "b86e85ab", "metadata": {}, "outputs": [ @@ -1977,7 +2082,7 @@ "true" ] }, - "execution_count": 89, + "execution_count": 94, "metadata": {}, "output_type": "execute_result" } @@ -1988,17 +2093,17 @@ }, { "cell_type": "code", - "execution_count": 90, + "execution_count": 95, "id": "8ed988ff", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ), Rows( Q ) ) ) supported on the interval [ 0 .. 2 ]>" + "GAP: " ] }, - "execution_count": 90, + "execution_count": 95, "metadata": {}, "output_type": "execute_result" } @@ -2009,7 +2114,7 @@ }, { "cell_type": "code", - "execution_count": 91, + "execution_count": 96, "id": "363eb7d6", "metadata": {}, "outputs": [ @@ -2078,17 +2183,17 @@ }, { "cell_type": "code", - "execution_count": 92, + "execution_count": 97, "id": "6d102b3a", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ), Rows( Q ) ) ) supported on the interval [ 0 .. 2 ]>" + "GAP: " ] }, - "execution_count": 92, + "execution_count": 97, "metadata": {}, "output_type": "execute_result" } @@ -2099,7 +2204,7 @@ }, { "cell_type": "code", - "execution_count": 93, + "execution_count": 98, "id": "670355fd", "metadata": {}, "outputs": [ @@ -2190,17 +2295,17 @@ }, { "cell_type": "code", - "execution_count": 94, + "execution_count": 99, "id": "7c9920ec", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ), Rows( Q ) ) ) supported on the interval [ 0 .. 2 ]>" + "GAP: " ] }, - "execution_count": 94, + "execution_count": 99, "metadata": {}, "output_type": "execute_result" } @@ -2211,7 +2316,7 @@ }, { "cell_type": "code", - "execution_count": 95, + "execution_count": 100, "id": "524ce9c0", "metadata": {}, "outputs": [ @@ -2221,7 +2326,7 @@ "true" ] }, - "execution_count": 95, + "execution_count": 100, "metadata": {}, "output_type": "execute_result" } @@ -2232,7 +2337,7 @@ }, { "cell_type": "code", - "execution_count": 96, + "execution_count": 101, "id": "6a91644d", "metadata": {}, "outputs": [ @@ -2290,7 +2395,7 @@ "\n", "A morphism in Rows( Q )\n", "\n", - "A morphism in PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ), Rows( Q ) ) given by the above data\n", + "A morphism in PreSheaves( Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms, Rows( Q ) ) given by the above data\n", "\n", "== 1 =======================\n", "Image of <(E1)>:\n", @@ -2348,7 +2453,7 @@ "\n", "A morphism in Rows( Q )\n", "\n", - "A morphism in PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ), Rows( Q ) ) given by the above data\n", + "A morphism in PreSheaves( Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms, Rows( Q ) ) given by the above data\n", "\n", "== 0 =======================\n", "Image of <(E1)>:\n", @@ -2401,10 +2506,10 @@ "\n", "A morphism in Rows( Q )\n", "\n", - "A morphism in PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ), Rows( Q ) ) given by the above data\n", + "A morphism in PreSheaves( Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms, Rows( Q ) ) given by the above data\n", "\n", "\n", - "A morphism in Homotopy category by cochains( PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ), Rows( Q ) ) ) defined by the above data\n" + "A morphism in Homotopy category by cochains( PreSheaves( Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms, Rows( Q ) ) ) defined by the above data\n" ] } ], @@ -2414,17 +2519,17 @@ }, { "cell_type": "code", - "execution_count": 97, + "execution_count": 102, "id": "0f50095c", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations, Rows( Q ) ) ) supported on the interval [ 0 .. 2 ]>" + "GAP: " ] }, - "execution_count": 97, + "execution_count": 102, "metadata": {}, "output_type": "execute_result" } @@ -2435,7 +2540,7 @@ }, { "cell_type": "code", - "execution_count": 98, + "execution_count": 103, "id": "c79361f0", "metadata": {}, "outputs": [ @@ -2444,7 +2549,7 @@ "text/latex": [ "$$\\begin{array}{c}\n", "\\begin{array}{ccc}\n", - " v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + " v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " 1 \n", "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{b} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " 1 \n", @@ -2457,19 +2562,19 @@ "\\uparrow_{\\phantom{1}}\n", "\\\\\n", "\\begin{array}{ccc}\n", - "v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + "v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " -1 \\\\ \n", " -1 \\\\ \n", " 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " 1 \\\\ \n", " -1 \\\\ \n", " 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " -1 \\\\ \n", " -1 \\\\ \n", " 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " -1 \\\\ \n", " 1 \n", "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\\\end{array}\n", @@ -2478,7 +2583,7 @@ "\n", "\\\\\n", "\\begin{array}{ccc}\n", - " v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 3} \\\\ v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 3} \\\\ v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 3} \\\\ v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 2} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrr}\n", + " v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 3} \\\\ v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 3} \\\\ v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 3} \\\\ v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 2} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrr}\n", " -1 & \\cdot & \\cdot \\\\ \n", " \\cdot & 1 & \\cdot \\\\ \n", " \\cdot & \\cdot & 1 \n", @@ -2496,17 +2601,17 @@ "\\uparrow_{\\phantom{0}}\n", "\\\\\n", "\\begin{array}{ccc}\n", - "v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrr}\n", + "v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrr}\n", " -1 & \\cdot & -1 \\\\ \n", " 1 & -1 & \\cdot \\\\ \n", " \\cdot & 1 & 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 3} \\\\ & & \\\\v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrr}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 3} \\\\ & & \\\\v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrr}\n", " 1 & \\cdot & -1 \\\\ \n", " \\cdot & 1 & 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 3} \\\\ & & \\\\v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrr}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 3} \\\\ & & \\\\v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrr}\n", " 1 & -1 & \\cdot \\\\ \n", " \\cdot & 1 & 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 3} \\\\ & & \\\\v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 3} \\\\ & & \\\\v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", " 1 & 1 \n", "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 2} \\\\ & & \\\\\\end{array}\n", "\\\\\n", @@ -2514,7 +2619,7 @@ "\n", "\\\\\n", "\\begin{array}{ccc}\n", - " v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 3} \\\\ v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 2} \\\\ v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 2} \\\\ v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrr}\n", + " v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 3} \\\\ v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 2} \\\\ v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 2} \\\\ v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrr}\n", " 1 & \\cdot & \\cdot \\\\ \n", " \\cdot & \\cdot & 1 \n", "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 3} \\\\ & & \\\\{b} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", @@ -2537,17 +2642,17 @@ }, { "cell_type": "code", - "execution_count": 99, + "execution_count": 104, "id": "641fd62c", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations, Rows( Q ) ) ) supported on the interval [ 0 .. 2 ]>" + "GAP: " ] }, - "execution_count": 99, + "execution_count": 104, "metadata": {}, "output_type": "execute_result" } @@ -2558,7 +2663,7 @@ }, { "cell_type": "code", - "execution_count": 100, + "execution_count": 105, "id": "921e07a0", "metadata": {}, "outputs": [ @@ -2568,7 +2673,7 @@ "true" ] }, - "execution_count": 100, + "execution_count": 105, "metadata": {}, "output_type": "execute_result" } @@ -2579,7 +2684,7 @@ }, { "cell_type": "code", - "execution_count": 101, + "execution_count": 106, "id": "ccec86e0", "metadata": {}, "outputs": [ @@ -2589,7 +2694,7 @@ "true" ] }, - "execution_count": 101, + "execution_count": 106, "metadata": {}, "output_type": "execute_result" } @@ -2600,7 +2705,7 @@ }, { "cell_type": "code", - "execution_count": 102, + "execution_count": 107, "id": "2b4c115a", "metadata": {}, "outputs": [ @@ -2610,7 +2715,7 @@ "true" ] }, - "execution_count": 102, + "execution_count": 107, "metadata": {}, "output_type": "execute_result" } @@ -2621,7 +2726,7 @@ }, { "cell_type": "code", - "execution_count": 103, + "execution_count": 108, "id": "e3b32c73", "metadata": {}, "outputs": [ @@ -2629,17 +2734,17 @@ "data": { "text/latex": [ "$$\\begin{array}{ccc}\n", - "v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + "v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " \\cdot \\\\ \n", " \\cdot \\\\ \n", " 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " \\cdot \\\\ \n", " 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " \\cdot \\\\ \n", " 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " 1 \n", "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\\\end{array}$$" ] @@ -2654,7 +2759,7 @@ }, { "cell_type": "code", - "execution_count": 104, + "execution_count": 109, "id": "947ecf2c", "metadata": {}, "outputs": [ @@ -2662,15 +2767,15 @@ "data": { "text/latex": [ "$$\\begin{array}{ccc}\n", - "v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{()_{3 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + "v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{()_{3 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " \\cdot \\\\ \n", " 1 \\\\ \n", " \\cdot \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " \\cdot \\\\ \n", " \\cdot \\\\ \n", " 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", " 1 & \\cdot \\\\ \n", " \\cdot & 1 \n", "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 2} \\\\ & & \\\\\\end{array}$$" @@ -2686,7 +2791,7 @@ }, { "cell_type": "code", - "execution_count": 105, + "execution_count": 110, "id": "3ff2924c", "metadata": {}, "outputs": [ @@ -2694,7 +2799,7 @@ "data": { "text/latex": [ "$$\\begin{array}{ccc}\n", - "v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{()_{1 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{()_{1 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{()_{1 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + "v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{()_{1 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{()_{1 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{()_{1 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " 1 \n", "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\\\end{array}$$" ] @@ -2709,17 +2814,17 @@ }, { "cell_type": "code", - "execution_count": 106, + "execution_count": 111, "id": "00d8fe3f", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ), Rows( Q ) ) ) supported on the interval [ 0 .. 2 ]>" + "GAP: " ] }, - "execution_count": 106, + "execution_count": 111, "metadata": {}, "output_type": "execute_result" } @@ -2730,17 +2835,17 @@ }, { "cell_type": "code", - "execution_count": 107, + "execution_count": 112, "id": "57680de1", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: Homotopy category by cochains( PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ), Rows( Q ) ) )" + "GAP: Homotopy category by cochains( PreSheaves( Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms, Rows( Q ) ) )" ] }, - "execution_count": 107, + "execution_count": 112, "metadata": {}, "output_type": "execute_result" } @@ -2751,7 +2856,7 @@ }, { "cell_type": "code", - "execution_count": 108, + "execution_count": 113, "id": "68a9b565", "metadata": {}, "outputs": [ @@ -2761,7 +2866,7 @@ "GAP: Localization functor via projective objects" ] }, - "execution_count": 108, + "execution_count": 113, "metadata": {}, "output_type": "execute_result" } @@ -2772,7 +2877,7 @@ }, { "cell_type": "code", - "execution_count": 109, + "execution_count": 114, "id": "13f330fc", "metadata": {}, "outputs": [ @@ -2782,10 +2887,10 @@ "text": [ "Localization functor via projective objects:\n", "\n", - "Homotopy category by cochains( PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ), Rows( Q ) ) )\n", + "Homotopy category by cochains( PreSheaves( Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms, Rows( Q ) ) )\n", " |\n", " V\n", - "Homotopy category by cochains( FullSubcategoryOfProjectiveObjects( PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ), Rows( Q ) ) ) )\n" + "Homotopy category by cochains( FullSubcategoryOfProjectiveObjects( PreSheaves( Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms, Rows( Q ) ) ) )\n" ] } ], @@ -2795,7 +2900,7 @@ }, { "cell_type": "code", - "execution_count": 110, + "execution_count": 115, "id": "0917ab5d", "metadata": {}, "outputs": [ @@ -2805,7 +2910,7 @@ "GAP: Extension of ( Equivalence functor from full subcategory of projective objects to source category ) to homotopy categories by cochains" ] }, - "execution_count": 110, + "execution_count": 115, "metadata": {}, "output_type": "execute_result" } @@ -2817,7 +2922,7 @@ }, { "cell_type": "code", - "execution_count": 111, + "execution_count": 116, "id": "dd3764c3", "metadata": {}, "outputs": [ @@ -2827,10 +2932,10 @@ "text": [ "Extension of ( Equivalence functor from full subcategory of projective objects to source category ) to homotopy categories by cochains:\n", "\n", - "Homotopy category by cochains( FullSubcategoryOfProjectiveObjects( PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ), Rows( Q ) ) ) )\n", + "Homotopy category by cochains( FullSubcategoryOfProjectiveObjects( PreSheaves( Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms, Rows( Q ) ) ) )\n", " |\n", " V\n", - "Homotopy category by cochains( AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ) ) )\n" + "Homotopy category by cochains( AdditiveClosure( Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms ) )\n" ] } ], @@ -2840,7 +2945,7 @@ }, { "cell_type": "code", - "execution_count": 112, + "execution_count": 117, "id": "1d6c9f37", "metadata": {}, "outputs": [ @@ -2850,7 +2955,7 @@ "GAP: Precomposition of Localization functor via projective objects and Extension of ( Equivalence functor from full subcategory of projective objects to source category ) to homotopy categories by cochains" ] }, - "execution_count": 112, + "execution_count": 117, "metadata": {}, "output_type": "execute_result" } @@ -2861,7 +2966,7 @@ }, { "cell_type": "code", - "execution_count": 113, + "execution_count": 118, "id": "0e461d84", "metadata": {}, "outputs": [ @@ -2871,10 +2976,10 @@ "text": [ "Precomposition of Localization functor via projective objects and Extension of ( Equivalence functor from full subcategory of projective objects to source category ) to homotopy categories by cochains:\n", "\n", - "Homotopy category by cochains( PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ), Rows( Q ) ) )\n", + "Homotopy category by cochains( PreSheaves( Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms, Rows( Q ) ) )\n", " |\n", " V\n", - "Homotopy category by cochains( AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ) ) )\n" + "Homotopy category by cochains( AdditiveClosure( Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms ) )\n" ] } ], @@ -2884,17 +2989,17 @@ }, { "cell_type": "code", - "execution_count": 114, + "execution_count": 119, "id": "382f31e5", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ) ) ) supported on the interval [ 0 .. 2 ]>" + "GAP: " ] }, - "execution_count": 114, + "execution_count": 119, "metadata": {}, "output_type": "execute_result" } @@ -2905,7 +3010,7 @@ }, { "cell_type": "code", - "execution_count": 115, + "execution_count": 120, "id": "cbff6339", "metadata": {}, "outputs": [ @@ -2918,8 +3023,8 @@ "\\uparrow_{\\phantom{1}}\n", "\\\\\n", "\\begin{pmatrix}-{m_{3,4}^{1}}\\\\ \n", - "-{E_{4}}\\\\ \n", - "{E_{4}}\\end{pmatrix}\n", + "-id(E_{4})\\\\ \n", + "id(E_{4})\\end{pmatrix}\n", "\\\\\n", "{\\vert_{1}}\n", "\n", @@ -2927,9 +3032,9 @@ "{E_{3}}\\oplus{E_{4}}^{\\oplus 2}\\\\\n", "\\uparrow_{\\phantom{0}}\n", "\\\\\n", - "\\begin{pmatrix}{m_{1,3}^{1}}&0&{m_{1,3}^{1}m_{3,4}^{1}}\\\\ \n", - "{m_{2,3}^{1}}&-{m_{2,3}^{1}m_{3,4}^{1}}&0\\\\ \n", - "0&{E_{4}}&{E_{4}}\\end{pmatrix}\n", + "\\begin{pmatrix}{m_{1,3}^{1}}&0&{m_{1,3}^{1}}{m_{3,4}^{1}}\\\\ \n", + "{m_{2,3}^{1}}&-{m_{2,3}^{1}}{m_{3,4}^{1}}&0\\\\ \n", + "0&id(E_{4})&id(E_{4})\\end{pmatrix}\n", "\\\\\n", "{\\vert_{0}}\n", "\n", @@ -2947,17 +3052,17 @@ }, { "cell_type": "code", - "execution_count": 116, + "execution_count": 121, "id": "3a511fc0", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: Homotopy category by cochains( AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ) ) )" + "GAP: Homotopy category by cochains( AdditiveClosure( Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms ) )" ] }, - "execution_count": 116, + "execution_count": 121, "metadata": {}, "output_type": "execute_result" } @@ -2968,17 +3073,17 @@ }, { "cell_type": "code", - "execution_count": 117, + "execution_count": 122, "id": "10da7f1e", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ) )" + "GAP: AdditiveClosure( Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms )" ] }, - "execution_count": 117, + "execution_count": 122, "metadata": {}, "output_type": "execute_result" } @@ -2989,17 +3094,17 @@ }, { "cell_type": "code", - "execution_count": 118, + "execution_count": 123, "id": "97ad3048", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ) ) ) supported on the interval [ 0 ]>" + "GAP: " ] }, - "execution_count": 118, + "execution_count": 123, "metadata": {}, "output_type": "execute_result" } @@ -3013,17 +3118,17 @@ }, { "cell_type": "code", - "execution_count": 119, + "execution_count": 124, "id": "f70af738", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: A strong exceptional sequence in Homotopy category by cochains( AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ) ) )" + "GAP: A strong exceptional sequence in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms ) )" ] }, - "execution_count": 119, + "execution_count": 124, "metadata": {}, "output_type": "execute_result" } @@ -3034,7 +3139,7 @@ }, { "cell_type": "code", - "execution_count": 120, + "execution_count": 125, "id": "4679a2a1", "metadata": {}, "outputs": [ @@ -3044,7 +3149,7 @@ "GAP: Convolution functor" ] }, - "execution_count": 120, + "execution_count": 125, "metadata": {}, "output_type": "execute_result" } @@ -3056,17 +3161,17 @@ }, { "cell_type": "code", - "execution_count": 121, + "execution_count": 126, "id": "c4c57b49", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ) ) ) supported on the interval [ 0 .. 1 ]>" + "GAP: " ] }, - "execution_count": 121, + "execution_count": 126, "metadata": {}, "output_type": "execute_result" } @@ -3077,7 +3182,7 @@ }, { "cell_type": "code", - "execution_count": 122, + "execution_count": 127, "id": "3c557f90", "metadata": {}, "outputs": [ @@ -3108,7 +3213,7 @@ }, { "cell_type": "code", - "execution_count": 123, + "execution_count": 128, "id": "a7f569d0", "metadata": {}, "outputs": [ @@ -3118,7 +3223,7 @@ "GAP: Counit ฯต : Fโˆ˜G โŸน Id of the adjunction F โŠฃ G" ] }, - "execution_count": 123, + "execution_count": 128, "metadata": {}, "output_type": "execute_result" } @@ -3129,17 +3234,17 @@ }, { "cell_type": "code", - "execution_count": 124, + "execution_count": 129, "id": "2cf392ee", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ) ) ) supported on the interval [ 0 .. 2 ]>" + "GAP: " ] }, - "execution_count": 124, + "execution_count": 129, "metadata": {}, "output_type": "execute_result" } @@ -3150,7 +3255,7 @@ }, { "cell_type": "code", - "execution_count": 125, + "execution_count": 130, "id": "34f98091", "metadata": {}, "outputs": [ @@ -3164,25 +3269,25 @@ " \\uparrow_{\\phantom{1}}\n", " \\\\ \n", " \\begin{pmatrix}\\\\\\end{pmatrix}&&\\begin{pmatrix}-{m_{3,4}^{1}}\\\\ \n", - "-{E_{4}}\\\\ \n", - "{E_{4}}\\end{pmatrix}\n", + "-id(E_{4})\\\\ \n", + "id(E_{4})\\end{pmatrix}\n", " \\\\ \n", " \\vert_{1} &&\\vert_{1} \n", " \\\\ \n", - " {E_{3}}&-\\phantom{-}{\\begin{pmatrix}{E_{3}}&-{m_{3,4}^{1}}&0\\end{pmatrix}}\\phantom{-}\\rightarrow&{E_{3}}\\oplus{E_{4}}^{\\oplus 2}\n", + " {E_{3}}&-\\phantom{-}{\\begin{pmatrix}id(E_{3})&-{m_{3,4}^{1}}&0\\end{pmatrix}}\\phantom{-}\\rightarrow&{E_{3}}\\oplus{E_{4}}^{\\oplus 2}\n", " \\\\ \n", " \\uparrow_{\\phantom{0}}&& \n", " \\uparrow_{\\phantom{0}}\n", " \\\\ \n", " \\begin{pmatrix}{m_{1,3}^{1}}\\\\ \n", - "-{m_{2,3}^{1}}\\end{pmatrix}&&\\begin{pmatrix}{m_{1,3}^{1}}&0&{m_{1,3}^{1}m_{3,4}^{1}}\\\\ \n", - "{m_{2,3}^{1}}&-{m_{2,3}^{1}m_{3,4}^{1}}&0\\\\ \n", - "0&{E_{4}}&{E_{4}}\\end{pmatrix}\n", + "-{m_{2,3}^{1}}\\end{pmatrix}&&\\begin{pmatrix}{m_{1,3}^{1}}&0&{m_{1,3}^{1}}{m_{3,4}^{1}}\\\\ \n", + "{m_{2,3}^{1}}&-{m_{2,3}^{1}}{m_{3,4}^{1}}&0\\\\ \n", + "0&id(E_{4})&id(E_{4})\\end{pmatrix}\n", " \\\\ \n", " \\vert_{0} &&\\vert_{0} \n", " \\\\ \n", - " {E_{1}}\\oplus{E_{2}}&-\\phantom{-}{\\begin{pmatrix}{E_{1}}&0&-{m_{1,3}^{1}m_{3,4}^{1}}\\\\ \n", - "0&-{E_{2}}&0\\end{pmatrix}}\\phantom{-}\\rightarrow&{E_{1}}\\oplus{E_{2}}\\oplus{E_{4}}\n", + " {E_{1}}\\oplus{E_{2}}&-\\phantom{-}{\\begin{pmatrix}id(E_{1})&0&-{m_{1,3}^{1}}{m_{3,4}^{1}}\\\\ \n", + "0&-id(E_{2})&0\\end{pmatrix}}\\phantom{-}\\rightarrow&{E_{1}}\\oplus{E_{2}}\\oplus{E_{4}}\n", " \\\\ \n", " \\end{array}$$" ] @@ -3197,7 +3302,7 @@ }, { "cell_type": "code", - "execution_count": 126, + "execution_count": 131, "id": "15edca80", "metadata": {}, "outputs": [ @@ -3207,7 +3312,7 @@ "true" ] }, - "execution_count": 126, + "execution_count": 131, "metadata": {}, "output_type": "execute_result" } @@ -3218,17 +3323,17 @@ }, { "cell_type": "code", - "execution_count": 127, + "execution_count": 132, "id": "ddd582b0", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ) ) ) supported on the interval [ 0 .. 2 ]>" + "GAP: " ] }, - "execution_count": 127, + "execution_count": 132, "metadata": {}, "output_type": "execute_result" } @@ -3239,7 +3344,7 @@ }, { "cell_type": "code", - "execution_count": 128, + "execution_count": 133, "id": "1c9f9dc2", "metadata": {}, "outputs": [ @@ -3253,27 +3358,27 @@ " \\uparrow_{\\phantom{1}}\n", " \\\\ \n", " \\begin{pmatrix}-{m_{3,4}^{1}}\\\\ \n", - "-{E_{4}}\\\\ \n", - "{E_{4}}\\end{pmatrix}&&\\begin{pmatrix}\\\\\\end{pmatrix}\n", + "-id(E_{4})\\\\ \n", + "id(E_{4})\\end{pmatrix}&&\\begin{pmatrix}\\\\\\end{pmatrix}\n", " \\\\ \n", " \\vert_{1} &&\\vert_{1} \n", " \\\\ \n", - " {E_{3}}\\oplus{E_{4}}^{\\oplus 2}&-\\phantom{-}{\\begin{pmatrix}{E_{3}}\\\\ \n", + " {E_{3}}\\oplus{E_{4}}^{\\oplus 2}&-\\phantom{-}{\\begin{pmatrix}id(E_{3})\\\\ \n", "0\\\\ \n", "0\\end{pmatrix}}\\phantom{-}\\rightarrow&{E_{3}}\n", " \\\\ \n", " \\uparrow_{\\phantom{0}}&& \n", " \\uparrow_{\\phantom{0}}\n", " \\\\ \n", - " \\begin{pmatrix}{m_{1,3}^{1}}&0&{m_{1,3}^{1}m_{3,4}^{1}}\\\\ \n", - "{m_{2,3}^{1}}&-{m_{2,3}^{1}m_{3,4}^{1}}&0\\\\ \n", - "0&{E_{4}}&{E_{4}}\\end{pmatrix}&&\\begin{pmatrix}{m_{1,3}^{1}}\\\\ \n", + " \\begin{pmatrix}{m_{1,3}^{1}}&0&{m_{1,3}^{1}}{m_{3,4}^{1}}\\\\ \n", + "{m_{2,3}^{1}}&-{m_{2,3}^{1}}{m_{3,4}^{1}}&0\\\\ \n", + "0&id(E_{4})&id(E_{4})\\end{pmatrix}&&\\begin{pmatrix}{m_{1,3}^{1}}\\\\ \n", "-{m_{2,3}^{1}}\\end{pmatrix}\n", " \\\\ \n", " \\vert_{0} &&\\vert_{0} \n", " \\\\ \n", - " {E_{1}}\\oplus{E_{2}}\\oplus{E_{4}}&-\\phantom{-}{\\begin{pmatrix}{E_{1}}&0\\\\ \n", - "0&-{E_{2}}\\\\ \n", + " {E_{1}}\\oplus{E_{2}}\\oplus{E_{4}}&-\\phantom{-}{\\begin{pmatrix}id(E_{1})&0\\\\ \n", + "0&-id(E_{2})\\\\ \n", "0&0\\end{pmatrix}}\\phantom{-}\\rightarrow&{E_{1}}\\oplus{E_{2}}\n", " \\\\ \n", " \\end{array}$$" @@ -3289,7 +3394,7 @@ }, { "cell_type": "code", - "execution_count": 129, + "execution_count": 134, "id": "23f6e1f0", "metadata": {}, "outputs": [ @@ -3299,7 +3404,7 @@ "GAP: Extension of ( Extension of Yoneda embedding functor to a functor from the additive closure of the source ) to homotopy categories by cochains" ] }, - "execution_count": 129, + "execution_count": 134, "metadata": {}, "output_type": "execute_result" } @@ -3312,7 +3417,7 @@ }, { "cell_type": "code", - "execution_count": 130, + "execution_count": 135, "id": "c1dbdc48", "metadata": {}, "outputs": [ @@ -3322,10 +3427,10 @@ "text": [ "Extension of ( Extension of Yoneda embedding functor to a functor from the additive closure of the source ) to homotopy categories by cochains:\n", "\n", - "Homotopy category by cochains( AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ) ) )\n", + "Homotopy category by cochains( AdditiveClosure( Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms ) )\n", " |\n", " V\n", - "Homotopy category by cochains( PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3,E4)[m1_3_1:E1->E3,m2_3_1:E2->E3,m3_4_1:E3->E4]\" ) ) ), Rows( Q ) ) )\n" + "Homotopy category by cochains( PreSheaves( Q-algebroid( {E1,E2,E3,E4}[m1_3_1:E1-โ‰ปE3,m2_3_1:E2-โ‰ปE3,m3_4_1:E3-โ‰ปE4] ) defined by 4 objects and 3 generating morphisms, Rows( Q ) ) )\n" ] } ], @@ -3335,17 +3440,17 @@ }, { "cell_type": "code", - "execution_count": 131, + "execution_count": 136, "id": "9405e758", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: v2,b:v2->v4,c:v1->v3,d:v3->v4]\" ) ) ) / relations, Rows( Q ) ) ) supported on the interval [ 0 .. 2 ]>" + "GAP: " ] }, - "execution_count": 131, + "execution_count": 136, "metadata": {}, "output_type": "execute_result" } @@ -3356,7 +3461,7 @@ }, { "cell_type": "code", - "execution_count": 132, + "execution_count": 137, "id": "3f529080", "metadata": {}, "outputs": [ @@ -3366,7 +3471,7 @@ "true" ] }, - "execution_count": 132, + "execution_count": 137, "metadata": {}, "output_type": "execute_result" } @@ -3377,7 +3482,7 @@ }, { "cell_type": "code", - "execution_count": 133, + "execution_count": 138, "id": "0e12a0c5", "metadata": {}, "outputs": [ @@ -3387,7 +3492,7 @@ "true" ] }, - "execution_count": 133, + "execution_count": 138, "metadata": {}, "output_type": "execute_result" } @@ -3398,7 +3503,7 @@ }, { "cell_type": "code", - "execution_count": 134, + "execution_count": 139, "id": "112453bb", "metadata": {}, "outputs": [ @@ -3407,7 +3512,7 @@ "text/latex": [ "$$\\begin{array}{c}\n", "\\begin{array}{ccc}\n", - " v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + " v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " -1 \n", "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{b} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{c} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " 1 \n", @@ -3416,20 +3521,20 @@ "\\uparrow_{\\phantom{0}}\n", "\\\\\n", "\\begin{array}{ccc}\n", - "v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + "v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 2}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " -1 \\\\ \n", " -1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{r}\n", " -1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\\\end{array}\n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 0}}}}\\mathbb{Q}^{1 \\times 0} \\\\ & & \\\\\\end{array}\n", "\\\\\n", "{\\vert_{0}}\n", "\n", "\\\\\n", "\\begin{array}{ccc}\n", - " v_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 2} \\\\ v_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_{4} & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", + " v_1 & \\mapsto & \\mathbb{Q}^{1 \\times 2} \\\\ v_2 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_3 & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ v_4 & \\mapsto & \\mathbb{Q}^{1 \\times 0} \\\\ \\hline & & \\\\{a} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", " 1 & \\cdot \n", "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 2} \\\\ & & \\\\{b} & \\mapsto & \\mathbb{Q}^{1 \\times 0}{\\color{blue}{\\xrightarrow{()_{0 \\times 1}}}}\\mathbb{Q}^{1 \\times 1} \\\\ & & \\\\{c} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rr}\n", " \\cdot & 1 \n", @@ -3443,14 +3548,6 @@ "source": [ "Show( Source( ฯ„ ) )" ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "fdbf288f", - "metadata": {}, - "outputs": [], - "source": [] } ], "metadata": { diff --git a/DerivedCategories/examples/notebooks/TiltingEquivalence.ipynb b/DerivedCategories/examples/notebooks/TiltingEquivalence.ipynb index 1d1e342b..84b14766 100644 --- a/DerivedCategories/examples/notebooks/TiltingEquivalence.ipynb +++ b/DerivedCategories/examples/notebooks/TiltingEquivalence.ipynb @@ -81,8 +81,8 @@ "metadata": {}, "outputs": [], "source": [ - "q_๐“ž = RightQuiver(\n", - " \"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€->๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" );" + "q_๐“ž = FinQuiver(\n", + " g\"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€->๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" );" ] }, { @@ -92,10 +92,8 @@ "metadata": {}, "outputs": [], "source": [ - "SetLabelsAsLaTeXStrings(\n", - " q_๐“ž,\n", - " [ \"\\\\mathcal{O}_{0}\", \"\\\\mathcal{O}_{1}\", \"\\\\mathcal{O}_{2}\" ],\n", - " [ \"x_0\", \"x_1\", \"x_2\", \"y_0\", \"y_1\", \"y_2\" ] )" + "SetLaTeXStringsOfObjects( q_๐“ž, ConvertJuliaToGAP([ \"\\\\mathcal{O}_{0}\", \"\\\\mathcal{O}_{1}\", \"\\\\mathcal{O}_{2}\" ]) );\n", + "SetLaTeXStringsOfMorphisms( q_๐“ž, ConvertJuliaToGAP([ \"x_0\", \"x_1\", \"x_2\", \"y_0\", \"y_1\", \"y_2\" ]) )" ] }, { @@ -107,7 +105,7 @@ { "data": { "text/plain": [ - "GAP: FreeCategory( RightQuiver( \"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€->๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) )" + "GAP: PathCategory( FinQuiver( \"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚‚:๐“žโ‚€-โ‰ป๐“žโ‚,yโ‚€:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚‚:๐“žโ‚-โ‰ป๐“žโ‚‚]\" ) )" ] }, "execution_count": 5, @@ -116,7 +114,7 @@ } ], "source": [ - "๐…_๐“ž = FreeCategory( q_๐“ž )" + "๐…_๐“ž = PathCategory( q_๐“ž )" ] }, { @@ -170,7 +168,7 @@ { "data": { "text/plain": [ - "GAP: Algebroid( Q, FreeCategory( RightQuiver( \"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€->๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) )" + "GAP: Q-LinearClosure( PathCategory( FinQuiver( \"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚‚:๐“žโ‚€-โ‰ป๐“žโ‚,yโ‚€:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚‚:๐“žโ‚-โ‰ป๐“žโ‚‚]\" ) ) )" ] }, "execution_count": 8, @@ -203,7 +201,7 @@ { "data": { "text/plain": [ - "GAP: Algebroid( Q, FreeCategory( RightQuiver( \"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€->๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations" + "GAP: Q-algebroid( {๐“žโ‚€,๐“žโ‚,๐“žโ‚‚}[xโ‚€:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚‚:๐“žโ‚€-โ‰ป๐“žโ‚,yโ‚€:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚‚:๐“žโ‚-โ‰ป๐“žโ‚‚] ) defined by 3 objects and 6 generating morphisms" ] }, "execution_count": 10, @@ -212,7 +210,7 @@ } ], "source": [ - "๐€_๐“ž = k๐…_๐“ž / ฯ_๐“ž" + "๐€_๐“ž = AlgebroidFromDataTables( k๐…_๐“ž / ฯ_๐“ž )" ] }, { @@ -227,9 +225,7 @@ "text": [ "27 primitive operations were used to derive 80 operations for this category which algorithmically\n", "* IsEquippedWithHomomorphismStructure\n", - "* IsLinearCategoryOverCommutativeRing\n", - "and furthermore mathematically\n", - "* IsFinitelyPresentedLinearCategory\n" + "* IsLinearCategoryOverCommutativeRing\n" ] } ], @@ -267,7 +263,7 @@ { "data": { "text/plain": [ - "GAP: (๐“žโ‚€)-[{ -1*(xโ‚‚) + 3*(xโ‚) + 2*(xโ‚€) }]->(๐“žโ‚)" + "GAP: <2*xโ‚€ + 3*xโ‚ - 1*xโ‚‚:(๐“žโ‚€) -โ‰ป (๐“žโ‚)>" ] }, "execution_count": 13, @@ -288,7 +284,7 @@ { "data": { "text/latex": [ - "$${\\mathcal{O}_{0}}-\\left({-{x_2}+3{x_1}+2{x_0}}\\right)\\rightarrow{\\mathcal{O}_{1}}$$" + "$${\\mathcal{O}_{0}}-\\left({2\\cdot {x_0} + 3\\cdot {x_1} - {x_2}}\\right)\\rightarrow{\\mathcal{O}_{1}}$$" ] }, "metadata": {}, @@ -342,7 +338,7 @@ "id": "b1657b55", "metadata": {}, "source": [ - "# The categories $\\mathbf{A}_{\\mathcal{O}}\\mbox{-}\\mathrm{mod},\\mathcal{K}^b(\\mathbf{A}_{\\mathcal{O}})$ and $\\mathcal{D}^b(\\mathbf{A}_{\\mathcal{O}})$" + "# The categories $\\mathrm{PSh}(\\mathbf{A}_{\\mathcal{O}}),\\mathcal{K}^b(\\mathbf{A}_{\\mathcal{O}})$ and $\\mathcal{D}^b(\\mathbf{A}_{\\mathcal{O}})$" ] }, { @@ -360,7 +356,7 @@ " $$\\mathbf{A}_\\mathcal{O} \\cong Y(\\mathbf{A}_\\mathcal{O}) \\mbox{.}$$\n", " \n", "For this $k$-linear Abelian functor category we use the notation\n", - "$$\\mathrm{PSh}(\\mathbf{A}_\\mathcal{O}) := \\mathrm{Hom}(\\mathbf{A}_\\mathcal{O}^\\mathrm{op},k\\mbox{-}\\mathbf{mat})$$\n", + "$$\\mathrm{PSh}(\\mathbf{A}_\\mathcal{O}) := \\mathrm{Hom}(\\mathbf{A}_\\mathcal{O}^\\mathrm{op},k\\mbox{-}\\mathbf{rows})$$\n", " \n", "and call it the $\\textbf{category finite $k$-dimensional $\\mathbf{A}_\\mathcal{O}$-presheaves}$." ] @@ -374,7 +370,7 @@ { "data": { "text/plain": [ - "GAP: Algebroid( Q, FreeCategory( RightQuiver( \"q_๐“ž_op(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚->๐“žโ‚€,xโ‚:๐“žโ‚->๐“žโ‚€,xโ‚‚:๐“žโ‚->๐“žโ‚€,yโ‚€:๐“žโ‚‚->๐“žโ‚,yโ‚:๐“žโ‚‚->๐“žโ‚,yโ‚‚:๐“žโ‚‚->๐“žโ‚]\" ) ) ) / relations" + "GAP: Q-algebroid( {๐“žโ‚€,๐“žโ‚,๐“žโ‚‚}[xโ‚€:๐“žโ‚-โ‰ป๐“žโ‚€,xโ‚:๐“žโ‚-โ‰ป๐“žโ‚€,xโ‚‚:๐“žโ‚-โ‰ป๐“žโ‚€,yโ‚€:๐“žโ‚‚-โ‰ป๐“žโ‚,yโ‚:๐“žโ‚‚-โ‰ป๐“žโ‚,yโ‚‚:๐“žโ‚‚-โ‰ป๐“žโ‚] ) defined by 3 objects and 6 generating morphisms" ] }, "execution_count": 16, @@ -395,7 +391,7 @@ { "data": { "text/plain": [ - "GAP: q_๐“ž_op(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚->๐“žโ‚€,xโ‚:๐“žโ‚->๐“žโ‚€,xโ‚‚:๐“žโ‚->๐“žโ‚€,yโ‚€:๐“žโ‚‚->๐“žโ‚,yโ‚:๐“žโ‚‚->๐“žโ‚,yโ‚‚:๐“žโ‚‚->๐“žโ‚]" + "GAP: FinQuiver( \"q_๐“ž_op(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚-โ‰ป๐“žโ‚€,xโ‚:๐“žโ‚-โ‰ป๐“žโ‚€,xโ‚‚:๐“žโ‚-โ‰ป๐“žโ‚€,yโ‚€:๐“žโ‚‚-โ‰ป๐“žโ‚,yโ‚:๐“žโ‚‚-โ‰ป๐“žโ‚,yโ‚‚:๐“žโ‚‚-โ‰ป๐“žโ‚]\" )" ] }, "execution_count": 17, @@ -410,29 +406,16 @@ { "cell_type": "code", "execution_count": 18, - "id": "2afe68b7", - "metadata": {}, - "outputs": [], - "source": [ - "SetLabelsAsLaTeXStrings(\n", - " q_๐“ž_op,\n", - " [ \"\\\\mathcal{O}(0)\", \"\\\\mathcal{O}(1)\", \"\\\\mathcal{O}(2)\" ],\n", - " [ \"x_0\", \"x_1\", \"x_2\", \"y_0\", \"y_1\", \"y_2\" ] )" - ] - }, - { - "cell_type": "code", - "execution_count": 19, "id": "d15108ee", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€->๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations, Rows( Q ) )" + "GAP: PreSheaves( Q-algebroid( {๐“žโ‚€,๐“žโ‚,๐“žโ‚‚}[xโ‚€:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚‚:๐“žโ‚€-โ‰ป๐“žโ‚,yโ‚€:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚‚:๐“žโ‚-โ‰ป๐“žโ‚‚] ) defined by 3 objects and 6 generating morphisms, Rows( Q ) )" ] }, - "execution_count": 19, + "execution_count": 18, "metadata": {}, "output_type": "execute_result" } @@ -443,7 +426,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 19, "id": "bd4bb462", "metadata": {}, "outputs": [ @@ -451,11 +434,10 @@ "name": "stdout", "output_type": "stream", "text": [ - "98 primitive operations were used to derive 347 operations for this category which algorithmically\n", + "92 primitive operations were used to derive 334 operations for this category which algorithmically\n", "* IsEquippedWithHomomorphismStructure\n", "* IsLinearCategoryOverCommutativeRing\n", - "* IsAbelianCategoryWithEnoughInjectives\n", - "* IsAbelianCategoryWithEnoughProjectives\n" + "* IsAbelianCategory\n" ] } ], @@ -485,17 +467,17 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 20, "id": "58074082", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: Homotopy category by cochains( PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€->๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations, Rows( Q ) ) )" + "GAP: Homotopy category by cochains( PreSheaves( Q-algebroid( {๐“žโ‚€,๐“žโ‚,๐“žโ‚‚}[xโ‚€:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚‚:๐“žโ‚€-โ‰ป๐“žโ‚,yโ‚€:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚‚:๐“žโ‚-โ‰ป๐“žโ‚‚] ) defined by 3 objects and 6 generating morphisms, Rows( Q ) ) )" ] }, - "execution_count": 21, + "execution_count": 20, "metadata": {}, "output_type": "execute_result" } @@ -506,17 +488,17 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 21, "id": "e9d9eb7d", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: Derived category by cochains( PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€->๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations, Rows( Q ) ) )" + "GAP: Derived category by cochains( PreSheaves( Q-algebroid( {๐“žโ‚€,๐“žโ‚,๐“žโ‚‚}[xโ‚€:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚‚:๐“žโ‚€-โ‰ป๐“žโ‚,yโ‚€:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚‚:๐“žโ‚-โ‰ป๐“žโ‚‚] ) defined by 3 objects and 6 generating morphisms, Rows( Q ) ) )" ] }, - "execution_count": 22, + "execution_count": 21, "metadata": {}, "output_type": "execute_result" } @@ -527,17 +509,17 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 22, "id": "bfdbf6ff", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "true" + "false" ] }, - "execution_count": 23, + "execution_count": 22, "metadata": {}, "output_type": "execute_result" } @@ -548,7 +530,7 @@ }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 23, "id": "55646dd7", "metadata": {}, "outputs": [ @@ -558,7 +540,7 @@ "GAP: Localization functor from homotopy category into derived category" ] }, - "execution_count": 24, + "execution_count": 23, "metadata": {}, "output_type": "execute_result" } @@ -569,7 +551,7 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 24, "id": "bc085c92", "metadata": {}, "outputs": [ @@ -579,10 +561,10 @@ "text": [ "Localization functor from homotopy category into derived category:\n", "\n", - "Homotopy category by cochains( PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€->๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations, Rows( Q ) ) )\n", + "Homotopy category by cochains( PreSheaves( Q-algebroid( {๐“žโ‚€,๐“žโ‚,๐“žโ‚‚}[xโ‚€:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚‚:๐“žโ‚€-โ‰ป๐“žโ‚,yโ‚€:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚‚:๐“žโ‚-โ‰ป๐“žโ‚‚] ) defined by 3 objects and 6 generating morphisms, Rows( Q ) ) )\n", " |\n", " V\n", - "Derived category by cochains( PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€->๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations, Rows( Q ) ) )\n" + "Derived category by cochains( PreSheaves( Q-algebroid( {๐“žโ‚€,๐“žโ‚,๐“žโ‚‚}[xโ‚€:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚‚:๐“žโ‚€-โ‰ป๐“žโ‚,yโ‚€:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚‚:๐“žโ‚-โ‰ป๐“žโ‚‚] ) defined by 3 objects and 6 generating morphisms, Rows( Q ) ) )\n" ] } ], @@ -614,17 +596,17 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 25, "id": "0a2c290b", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€->๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations )" + "GAP: AdditiveClosure( Q-algebroid( {๐“žโ‚€,๐“žโ‚,๐“žโ‚‚}[xโ‚€:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚‚:๐“žโ‚€-โ‰ป๐“žโ‚,yโ‚€:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚‚:๐“žโ‚-โ‰ป๐“žโ‚‚] ) defined by 3 objects and 6 generating morphisms )" ] }, - "execution_count": 26, + "execution_count": 25, "metadata": {}, "output_type": "execute_result" } @@ -635,17 +617,17 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 26, "id": "6bab702a", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: Homotopy category by cochains( AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€->๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) )" + "GAP: Homotopy category by cochains( AdditiveClosure( Q-algebroid( {๐“žโ‚€,๐“žโ‚,๐“žโ‚‚}[xโ‚€:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚‚:๐“žโ‚€-โ‰ป๐“žโ‚,yโ‚€:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚‚:๐“žโ‚-โ‰ป๐“žโ‚‚] ) defined by 3 objects and 6 generating morphisms ) )" ] }, - "execution_count": 27, + "execution_count": 26, "metadata": {}, "output_type": "execute_result" } @@ -656,7 +638,7 @@ }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 27, "id": "d0030a78", "metadata": {}, "outputs": [ @@ -666,7 +648,7 @@ "GAP: Equivalence functor onto derived category of presheaves" ] }, - "execution_count": 28, + "execution_count": 27, "metadata": {}, "output_type": "execute_result" } @@ -677,7 +659,7 @@ }, { "cell_type": "code", - "execution_count": 29, + "execution_count": 28, "id": "77cc7979", "metadata": {}, "outputs": [ @@ -687,10 +669,10 @@ "text": [ "Equivalence functor onto derived category of presheaves:\n", "\n", - "Homotopy category by cochains( AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€->๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) )\n", + "Homotopy category by cochains( AdditiveClosure( Q-algebroid( {๐“žโ‚€,๐“žโ‚,๐“žโ‚‚}[xโ‚€:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚‚:๐“žโ‚€-โ‰ป๐“žโ‚,yโ‚€:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚‚:๐“žโ‚-โ‰ป๐“žโ‚‚] ) defined by 3 objects and 6 generating morphisms ) )\n", " |\n", " V\n", - "Derived category by cochains( PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€->๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations, Rows( Q ) ) )\n" + "Derived category by cochains( PreSheaves( Q-algebroid( {๐“žโ‚€,๐“žโ‚,๐“žโ‚‚}[xโ‚€:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚‚:๐“žโ‚€-โ‰ป๐“žโ‚,yโ‚€:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚‚:๐“žโ‚-โ‰ป๐“žโ‚‚] ) defined by 3 objects and 6 generating morphisms, Rows( Q ) ) )\n" ] } ], @@ -700,17 +682,17 @@ }, { "cell_type": "code", - "execution_count": 30, + "execution_count": 29, "id": "0a0dacc3", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: ๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) ) supported on the interval [ -4 .. 4 ]>" + "GAP: " ] }, - "execution_count": 30, + "execution_count": 29, "metadata": {}, "output_type": "execute_result" } @@ -721,7 +703,7 @@ }, { "cell_type": "code", - "execution_count": 31, + "execution_count": 30, "id": "46d08f67", "metadata": {}, "outputs": [], @@ -731,17 +713,17 @@ }, { "cell_type": "code", - "execution_count": 32, + "execution_count": 31, "id": "706596bf", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: ๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations, Rows( Q ) ) ) supported on the interval [ -4 .. 4 ]>" + "GAP: " ] }, - "execution_count": 32, + "execution_count": 31, "metadata": {}, "output_type": "execute_result" } @@ -752,7 +734,7 @@ }, { "cell_type": "code", - "execution_count": 33, + "execution_count": 32, "id": "3bdde051", "metadata": { "scrolled": true @@ -787,17 +769,17 @@ }, { "cell_type": "code", - "execution_count": 34, + "execution_count": 33, "id": "947bc816", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: ๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) defined by 1 underlying objects>" + "GAP: " ] }, - "execution_count": 34, + "execution_count": 33, "metadata": {}, "output_type": "execute_result" } @@ -810,7 +792,7 @@ }, { "cell_type": "code", - "execution_count": 35, + "execution_count": 34, "id": "729b0c41", "metadata": {}, "outputs": [ @@ -830,17 +812,17 @@ }, { "cell_type": "code", - "execution_count": 36, + "execution_count": 35, "id": "74283f98", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: ๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) defined by a 3 x 1 matrix of underlying morphisms>" + "GAP: " ] }, - "execution_count": 36, + "execution_count": 35, "metadata": {}, "output_type": "execute_result" } @@ -865,7 +847,7 @@ }, { "cell_type": "code", - "execution_count": 37, + "execution_count": 36, "id": "b5d70c08", "metadata": {}, "outputs": [ @@ -887,17 +869,17 @@ }, { "cell_type": "code", - "execution_count": 38, + "execution_count": 37, "id": "1691e613", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: ๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) ) supported on the interval [ 0 .. 2 ]>" + "GAP: " ] }, - "execution_count": 38, + "execution_count": 37, "metadata": {}, "output_type": "execute_result" } @@ -908,7 +890,7 @@ }, { "cell_type": "code", - "execution_count": 39, + "execution_count": 38, "id": "f79b46aa", "metadata": {}, "outputs": [ @@ -950,7 +932,7 @@ }, { "cell_type": "code", - "execution_count": 40, + "execution_count": 39, "id": "36724977", "metadata": {}, "outputs": [ @@ -960,7 +942,7 @@ "true" ] }, - "execution_count": 40, + "execution_count": 39, "metadata": {}, "output_type": "execute_result" } @@ -971,7 +953,7 @@ }, { "cell_type": "code", - "execution_count": 41, + "execution_count": 40, "id": "bbddcd67", "metadata": {}, "outputs": [], @@ -982,17 +964,17 @@ }, { "cell_type": "code", - "execution_count": 42, + "execution_count": 41, "id": "cc4cb406", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: ๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) defined by a 3 x 1 matrix of underlying morphisms>" + "GAP: " ] }, - "execution_count": 42, + "execution_count": 41, "metadata": {}, "output_type": "execute_result" } @@ -1009,17 +991,17 @@ }, { "cell_type": "code", - "execution_count": 43, + "execution_count": 42, "id": "6778cc17", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: ๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) ) supported on the interval [ 0 .. 1 ]>" + "GAP: " ] }, - "execution_count": 43, + "execution_count": 42, "metadata": {}, "output_type": "execute_result" } @@ -1030,7 +1012,7 @@ }, { "cell_type": "code", - "execution_count": 44, + "execution_count": 43, "id": "4d6572ef", "metadata": {}, "outputs": [ @@ -1062,17 +1044,17 @@ }, { "cell_type": "code", - "execution_count": 45, + "execution_count": 44, "id": "3ff405d3", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: ๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) ) supported on the interval [ 0 ]>" + "GAP: " ] }, - "execution_count": 45, + "execution_count": 44, "metadata": {}, "output_type": "execute_result" } @@ -1083,7 +1065,7 @@ }, { "cell_type": "code", - "execution_count": 46, + "execution_count": 45, "id": "fb076a15", "metadata": {}, "outputs": [ @@ -1105,17 +1087,17 @@ }, { "cell_type": "code", - "execution_count": 47, + "execution_count": 46, "id": "a126b2a5", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: A strong exceptional sequence in Homotopy category by cochains( AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€->๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) )" + "GAP: A strong exceptional sequence in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {๐“žโ‚€,๐“žโ‚,๐“žโ‚‚}[xโ‚€:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚‚:๐“žโ‚€-โ‰ป๐“žโ‚,yโ‚€:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚‚:๐“žโ‚-โ‰ป๐“žโ‚‚] ) defined by 3 objects and 6 generating morphisms ) )" ] }, - "execution_count": 47, + "execution_count": 46, "metadata": {}, "output_type": "execute_result" } @@ -1126,17 +1108,17 @@ }, { "cell_type": "code", - "execution_count": 48, + "execution_count": 47, "id": "aea2dd2c", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: ๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) ) supported on the interval [ 0 .. 2 ]>" + "GAP: " ] }, - "execution_count": 48, + "execution_count": 47, "metadata": {}, "output_type": "execute_result" } @@ -1147,7 +1129,7 @@ }, { "cell_type": "code", - "execution_count": 49, + "execution_count": 48, "id": "959d8e2f", "metadata": {}, "outputs": [ @@ -1202,7 +1184,7 @@ }, { "cell_type": "code", - "execution_count": 50, + "execution_count": 49, "id": "68a39658", "metadata": {}, "outputs": [ @@ -1212,7 +1194,7 @@ "true" ] }, - "execution_count": 50, + "execution_count": 49, "metadata": {}, "output_type": "execute_result" } @@ -1225,7 +1207,7 @@ }, { "cell_type": "code", - "execution_count": 51, + "execution_count": 50, "id": "bbc10e3b", "metadata": {}, "outputs": [ @@ -1235,7 +1217,7 @@ "true" ] }, - "execution_count": 51, + "execution_count": 50, "metadata": {}, "output_type": "execute_result" } @@ -1248,7 +1230,7 @@ }, { "cell_type": "code", - "execution_count": 52, + "execution_count": 51, "id": "187cce3e", "metadata": {}, "outputs": [ @@ -1258,7 +1240,7 @@ "true" ] }, - "execution_count": 52, + "execution_count": 51, "metadata": {}, "output_type": "execute_result" } @@ -1272,7 +1254,7 @@ }, { "cell_type": "code", - "execution_count": 53, + "execution_count": 52, "id": "143cf183", "metadata": {}, "outputs": [ @@ -1282,7 +1264,7 @@ "GAP: " ] }, - "execution_count": 53, + "execution_count": 52, "metadata": {}, "output_type": "execute_result" } @@ -1323,17 +1305,17 @@ }, { "cell_type": "code", - "execution_count": 54, + "execution_count": 53, "id": "7874f727", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: ๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations, Rows( Q ) ) ) supported on the interval [ 0 .. 2 ]>" + "GAP: " ] }, - "execution_count": 54, + "execution_count": 53, "metadata": {}, "output_type": "execute_result" } @@ -1344,7 +1326,7 @@ }, { "cell_type": "code", - "execution_count": 55, + "execution_count": 54, "id": "e306ec1b", "metadata": {}, "outputs": [ @@ -1354,7 +1336,7 @@ "GAP: [ 2 ]" ] }, - "execution_count": 55, + "execution_count": 54, "metadata": {}, "output_type": "execute_result" } @@ -1365,7 +1347,7 @@ }, { "cell_type": "code", - "execution_count": 56, + "execution_count": 55, "id": "11995f6e", "metadata": {}, "outputs": [ @@ -1375,7 +1357,7 @@ "GAP: <(๐“žโ‚€)->0, (๐“žโ‚)->0, (๐“žโ‚‚)->1; (xโ‚€)->0x0, (xโ‚)->0x0, (xโ‚‚)->0x0, (yโ‚€)->1x0, (yโ‚)->1x0, (yโ‚‚)->1x0>" ] }, - "execution_count": 56, + "execution_count": 55, "metadata": {}, "output_type": "execute_result" } @@ -1386,7 +1368,7 @@ }, { "cell_type": "code", - "execution_count": 57, + "execution_count": 56, "id": "8cebf4ab", "metadata": {}, "outputs": [ @@ -1407,17 +1389,17 @@ }, { "cell_type": "code", - "execution_count": 58, + "execution_count": 57, "id": "b5118f72", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: ๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations, Rows( Q ) ) ) supported on the interval [ 0 .. 1 ]>" + "GAP: " ] }, - "execution_count": 58, + "execution_count": 57, "metadata": {}, "output_type": "execute_result" } @@ -1428,7 +1410,7 @@ }, { "cell_type": "code", - "execution_count": 59, + "execution_count": 58, "id": "70ca8459", "metadata": {}, "outputs": [ @@ -1438,7 +1420,7 @@ "GAP: [ 1 ]" ] }, - "execution_count": 59, + "execution_count": 58, "metadata": {}, "output_type": "execute_result" } @@ -1449,7 +1431,7 @@ }, { "cell_type": "code", - "execution_count": 60, + "execution_count": 59, "id": "3bc559ba", "metadata": {}, "outputs": [ @@ -1459,7 +1441,7 @@ "GAP: <(๐“žโ‚€)->0, (๐“žโ‚)->1, (๐“žโ‚‚)->0; (xโ‚€)->1x0, (xโ‚)->1x0, (xโ‚‚)->1x0, (yโ‚€)->0x1, (yโ‚)->0x1, (yโ‚‚)->0x1>" ] }, - "execution_count": 60, + "execution_count": 59, "metadata": {}, "output_type": "execute_result" } @@ -1470,7 +1452,7 @@ }, { "cell_type": "code", - "execution_count": 61, + "execution_count": 60, "id": "5c2a152f", "metadata": {}, "outputs": [ @@ -1491,17 +1473,17 @@ }, { "cell_type": "code", - "execution_count": 62, + "execution_count": 61, "id": "918e6cee", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: ๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations, Rows( Q ) ) ) supported on the interval [ 0 ]>" + "GAP: " ] }, - "execution_count": 62, + "execution_count": 61, "metadata": {}, "output_type": "execute_result" } @@ -1512,7 +1494,7 @@ }, { "cell_type": "code", - "execution_count": 63, + "execution_count": 62, "id": "f611dc3c", "metadata": {}, "outputs": [ @@ -1522,7 +1504,7 @@ "GAP: [ 0 ]" ] }, - "execution_count": 63, + "execution_count": 62, "metadata": {}, "output_type": "execute_result" } @@ -1533,7 +1515,7 @@ }, { "cell_type": "code", - "execution_count": 64, + "execution_count": 63, "id": "d5233f29", "metadata": {}, "outputs": [ @@ -1543,7 +1525,7 @@ "GAP: <(๐“žโ‚€)->1, (๐“žโ‚)->0, (๐“žโ‚‚)->0; (xโ‚€)->0x1, (xโ‚)->0x1, (xโ‚‚)->0x1, (yโ‚€)->0x0, (yโ‚)->0x0, (yโ‚‚)->0x0>" ] }, - "execution_count": 64, + "execution_count": 63, "metadata": {}, "output_type": "execute_result" } @@ -1554,7 +1536,7 @@ }, { "cell_type": "code", - "execution_count": 65, + "execution_count": 64, "id": "1a375cc1", "metadata": {}, "outputs": [ @@ -1593,17 +1575,17 @@ }, { "cell_type": "code", - "execution_count": 66, + "execution_count": 65, "id": "c4334c50", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3)[m1_2_1:E1->E2,m1_2_2:E1->E2,m1_2_3:E1->E2,m2_3_1:E2->E3,m2_3_2:E2->E3,m2_3_3:E2->E3]\" ) ) ) / relations" + "GAP: Q-algebroid( {E1,E2,E3}[m1_2_1:E1-โ‰ปE2,m1_2_2:E1-โ‰ปE2,m1_2_3:E1-โ‰ปE2,m2_3_1:E2-โ‰ปE3,m2_3_2:E2-โ‰ปE3,m2_3_3:E2-โ‰ปE3] ) defined by 3 objects and 6 generating morphisms" ] }, - "execution_count": 66, + "execution_count": 65, "metadata": {}, "output_type": "execute_result" } @@ -1614,17 +1596,17 @@ }, { "cell_type": "code", - "execution_count": 67, + "execution_count": 66, "id": "c0e12a69", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: q(E1,E2,E3)[m1_2_1:E1->E2,m1_2_2:E1->E2,m1_2_3:E1->E2,m2_3_1:E2->E3,m2_3_2:E2->E3,m2_3_3:E2->E3]" + "GAP: FinQuiver( \"q(E1,E2,E3)[m1_2_1:E1-โ‰ปE2,m1_2_2:E1-โ‰ปE2,m1_2_3:E1-โ‰ปE2,m2_3_1:E2-โ‰ปE3,m2_3_2:E2-โ‰ปE3,m2_3_3:E2-โ‰ปE3]\" )" ] }, - "execution_count": 67, + "execution_count": 66, "metadata": {}, "output_type": "execute_result" } @@ -1635,65 +1617,65 @@ }, { "cell_type": "code", - "execution_count": 68, - "id": "0b2e966c", + "execution_count": 67, + "id": "48b88e07", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: (Q * q) / [ 1*(m1_2_1*m2_3_1), 1*(m1_2_2*m2_3_1) + 1*(m1_2_1*m2_3_2), 1*(m1_2_2*m2_3_2), 1*(m1_2_3*m2_3_1) + 1*(m1_2_1*m2_3_3), 1*(m1_2_3*m2_3_2) + 1*(m1_2_2*m2_3_3), 1*(m1_2_3*m2_3_3) ]" + "12" ] }, - "execution_count": 68, + "execution_count": 67, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "B_๐“” = UnderlyingQuiverAlgebra( ๐€_๐“” )" + "Dimension( ๐€_๐“” )" ] }, { "cell_type": "code", - "execution_count": 69, - "id": "48b88e07", + "execution_count": 68, + "id": "d72e479a-25c8-4162-be05-0daa638aec13", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "12" + "GAP: Q-LinearClosure( PathCategory( FinQuiver( \"q(E1,E2,E3)[m1_2_1:E1-โ‰ปE2,m1_2_2:E1-โ‰ปE2,m1_2_3:E1-โ‰ปE2,m2_3_1:E2-โ‰ปE3,m2_3_2:E2-โ‰ปE3,m2_3_3:E2-โ‰ปE3]\" ) ) ) / [ 1*m1_2_1โ€ขm2_3_1, 1*m1_2_1โ€ขm2_3_2 + 1*m1_2_2โ€ขm2_3_1, 1*m1_2_2โ€ขm2_3_2, ... ]" ] }, - "execution_count": 69, + "execution_count": 68, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "Dimension( B_๐“” )" + "quo_C = DefiningCategory( ๐€_๐“” )" ] }, { "cell_type": "code", - "execution_count": 70, + "execution_count": 69, "id": "90993dc2", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: [ (E1)-[1*(m1_2_1*m2_3_1)]->(E3), (E1)-[1*(m1_2_2*m2_3_1) + 1*(m1_2_1*m2_3_2)]->(E3), (E1)-[1*(m1_2_2*m2_3_2)]->(E3), (E1)-[1*(m1_2_3*m2_3_1) + 1*(m1_2_1*m2_3_3)]->(E3), (E1)-[1*(m1_2_3*m2_3_2) + 1*(m1_2_2*m2_3_3)]->(E3), (E1)-[1*(m1_2_3*m2_3_3)]->(E3) ]" + "GAP: [ 1*m1_2_1โ€ขm2_3_1:(E1) -โ‰ป (E3), 1*m1_2_1โ€ขm2_3_2 + 1*m1_2_2โ€ขm2_3_1:(E1) -โ‰ป (E3), 1*m1_2_2โ€ขm2_3_2:(E1) -โ‰ป (E3), 1*m1_2_1โ€ขm2_3_3 + 1*m1_2_3โ€ขm2_3_1:(E1) -โ‰ป (E3), 1*m1_2_2โ€ขm2_3_3 + 1*m1_2_3โ€ขm2_3_2:(E1) -โ‰ป (E3), 1*m1_2_3โ€ขm2_3_3:(E1) -โ‰ป (E3) ]" ] }, - "execution_count": 70, + "execution_count": 69, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "ฯ_๐“” = RelationsOfAlgebroid( ๐€_๐“” )" + "ฯ_๐“” = DefiningRelations( quo_C )" ] }, { @@ -1714,7 +1696,7 @@ }, { "cell_type": "code", - "execution_count": 71, + "execution_count": 70, "id": "a5217bf9", "metadata": {}, "outputs": [ @@ -1724,7 +1706,7 @@ "GAP: Isomorphism: strong exceptional sequence โŸถ abstraction algebroid" ] }, - "execution_count": 71, + "execution_count": 70, "metadata": {}, "output_type": "execute_result" } @@ -1735,7 +1717,7 @@ }, { "cell_type": "code", - "execution_count": 72, + "execution_count": 71, "id": "ed13dceb", "metadata": {}, "outputs": [ @@ -1745,10 +1727,10 @@ "text": [ "Isomorphism: strong exceptional sequence โŸถ abstraction algebroid:\n", "\n", - "A strong exceptional sequence in Homotopy category by cochains( AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€->๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) )\n", + "A strong exceptional sequence in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {๐“žโ‚€,๐“žโ‚,๐“žโ‚‚}[xโ‚€:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚‚:๐“žโ‚€-โ‰ป๐“žโ‚,yโ‚€:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚‚:๐“žโ‚-โ‰ป๐“žโ‚‚] ) defined by 3 objects and 6 generating morphisms ) )\n", " |\n", " V\n", - "Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3)[m1_2_1:E1->E2,m1_2_2:E1->E2,m1_2_3:E1->E2,m2_3_1:E2->E3,m2_3_2:E2->E3,m2_3_3:E2->E3]\" ) ) ) / relations\n" + "Q-algebroid( {E1,E2,E3}[m1_2_1:E1-โ‰ปE2,m1_2_2:E1-โ‰ปE2,m1_2_3:E1-โ‰ปE2,m2_3_1:E2-โ‰ปE3,m2_3_2:E2-โ‰ปE3,m2_3_3:E2-โ‰ปE3] ) defined by 3 objects and 6 generating morphisms\n" ] } ], @@ -1758,7 +1740,7 @@ }, { "cell_type": "code", - "execution_count": 73, + "execution_count": 72, "id": "7f64dfd9", "metadata": {}, "outputs": [ @@ -1768,7 +1750,7 @@ "GAP: Isomorphism: abstraction algebroid โŸถ strong exceptional sequence" ] }, - "execution_count": 73, + "execution_count": 72, "metadata": {}, "output_type": "execute_result" } @@ -1779,17 +1761,17 @@ }, { "cell_type": "code", - "execution_count": 74, + "execution_count": 73, "id": "40032394", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: (E1)-[{ 1*(m1_2_1) }]->(E2)" + "GAP: <1*m1_2_1:(E1) -โ‰ป (E2)>" ] }, - "execution_count": 74, + "execution_count": 73, "metadata": {}, "output_type": "execute_result" } @@ -1800,17 +1782,17 @@ }, { "cell_type": "code", - "execution_count": 75, + "execution_count": 74, "id": "d1a979ea", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: A morphism in full subcategory given by: ๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) ) supported on the interval [ 0 .. 2 ]>" + "GAP: A morphism in full subcategory given by: " ] }, - "execution_count": 75, + "execution_count": 74, "metadata": {}, "output_type": "execute_result" } @@ -1821,7 +1803,7 @@ }, { "cell_type": "code", - "execution_count": 76, + "execution_count": 75, "id": "f26a41cc", "metadata": {}, "outputs": [ @@ -1840,7 +1822,7 @@ " \\\\ \n", " \\vert_{1} &&\\vert_{1} \n", " \\\\ \n", - " {\\mathcal{O}_{1}}^{\\oplus 3}&-\\phantom{-}{\\begin{pmatrix}{\\mathcal{O}_{1}}\\\\ \n", + " {\\mathcal{O}_{1}}^{\\oplus 3}&-\\phantom{-}{\\begin{pmatrix}id(\\mathcal{O}_{1})\\\\ \n", "0\\\\ \n", "0\\end{pmatrix}}\\phantom{-}\\rightarrow&{\\mathcal{O}_{1}}\n", " \\\\ \n", @@ -1855,8 +1837,8 @@ " \\\\ \n", " \\vert_{0} &&\\vert_{0} \n", " \\\\ \n", - " {\\mathcal{O}_{0}}^{\\oplus 3}&-\\phantom{-}{\\begin{pmatrix}0&{\\mathcal{O}_{0}}&0\\\\ \n", - "0&0&{\\mathcal{O}_{0}}\\\\ \n", + " {\\mathcal{O}_{0}}^{\\oplus 3}&-\\phantom{-}{\\begin{pmatrix}0&id(\\mathcal{O}_{0})&0\\\\ \n", + "0&0&id(\\mathcal{O}_{0})\\\\ \n", "0&0&0\\end{pmatrix}}\\phantom{-}\\rightarrow&{\\mathcal{O}_{0}}^{\\oplus 3}\n", " \\\\ \n", " \\end{array}$$" @@ -1872,7 +1854,7 @@ }, { "cell_type": "code", - "execution_count": 77, + "execution_count": 76, "id": "ee401e1a", "metadata": {}, "outputs": [ @@ -1882,7 +1864,7 @@ "true" ] }, - "execution_count": 77, + "execution_count": 76, "metadata": {}, "output_type": "execute_result" } @@ -1913,17 +1895,17 @@ }, { "cell_type": "code", - "execution_count": 78, + "execution_count": 77, "id": "5579b474", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: TriangulatedSubcategory( A strong exceptional sequence in Homotopy category by cochains( AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€->๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) ) )" + "GAP: TriangulatedSubcategory( A strong exceptional sequence in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {๐“žโ‚€,๐“žโ‚,๐“žโ‚‚}[xโ‚€:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚‚:๐“žโ‚€-โ‰ป๐“žโ‚,yโ‚€:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚‚:๐“žโ‚-โ‰ป๐“žโ‚‚] ) defined by 3 objects and 6 generating morphisms ) ) )" ] }, - "execution_count": 78, + "execution_count": 77, "metadata": {}, "output_type": "execute_result" } @@ -1934,17 +1916,17 @@ }, { "cell_type": "code", - "execution_count": 79, + "execution_count": 78, "id": "9d4e6154", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: ๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) ) supported on the interval [ 0 ]>" + "GAP: " ] }, - "execution_count": 79, + "execution_count": 78, "metadata": {}, "output_type": "execute_result" } @@ -1955,17 +1937,17 @@ }, { "cell_type": "code", - "execution_count": 80, + "execution_count": 79, "id": "6697409f", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: ๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) ) supported on the interval [ 0 ]>" + "GAP: " ] }, - "execution_count": 80, + "execution_count": 79, "metadata": {}, "output_type": "execute_result" } @@ -1976,17 +1958,17 @@ }, { "cell_type": "code", - "execution_count": 81, + "execution_count": 80, "id": "3cca77c4", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: ๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) ) supported on the interval [ 0 ]>" + "GAP: " ] }, - "execution_count": 81, + "execution_count": 80, "metadata": {}, "output_type": "execute_result" } @@ -1997,7 +1979,7 @@ }, { "cell_type": "code", - "execution_count": 82, + "execution_count": 81, "id": "873cdb0d", "metadata": {}, "outputs": [ @@ -2007,7 +1989,7 @@ "true" ] }, - "execution_count": 82, + "execution_count": 81, "metadata": {}, "output_type": "execute_result" } @@ -2036,7 +2018,7 @@ }, { "cell_type": "code", - "execution_count": 83, + "execution_count": 82, "id": "a66b9c0e", "metadata": {}, "outputs": [ @@ -2046,7 +2028,7 @@ "GAP: Replacement functor" ] }, - "execution_count": 83, + "execution_count": 82, "metadata": {}, "output_type": "execute_result" } @@ -2057,7 +2039,7 @@ }, { "cell_type": "code", - "execution_count": 84, + "execution_count": 83, "id": "cd848fe6", "metadata": {}, "outputs": [ @@ -2067,10 +2049,10 @@ "text": [ "Replacement functor:\n", "\n", - "Homotopy category by cochains( AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q_๐“ž(๐“žโ‚€,๐“žโ‚,๐“žโ‚‚)[xโ‚€:๐“žโ‚€->๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) )\n", + "Homotopy category by cochains( AdditiveClosure( Q-algebroid( {๐“žโ‚€,๐“žโ‚,๐“žโ‚‚}[xโ‚€:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚:๐“žโ‚€-โ‰ป๐“žโ‚,xโ‚‚:๐“žโ‚€-โ‰ป๐“žโ‚,yโ‚€:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚:๐“žโ‚-โ‰ป๐“žโ‚‚,yโ‚‚:๐“žโ‚-โ‰ป๐“žโ‚‚] ) defined by 3 objects and 6 generating morphisms ) )\n", " |\n", " V\n", - "Homotopy category by cochains( AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3)[m1_2_1:E1->E2,m1_2_2:E1->E2,m1_2_3:E1->E2,m2_3_1:E2->E3,m2_3_2:E2->E3,m2_3_3:E2->E3]\" ) ) ) / relations ) )\n" + "Homotopy category by cochains( AdditiveClosure( Q-algebroid( {E1,E2,E3}[m1_2_1:E1-โ‰ปE2,m1_2_2:E1-โ‰ปE2,m1_2_3:E1-โ‰ปE2,m2_3_1:E2-โ‰ปE3,m2_3_2:E2-โ‰ปE3,m2_3_3:E2-โ‰ปE3] ) defined by 3 objects and 6 generating morphisms ) )\n" ] } ], @@ -2080,7 +2062,7 @@ }, { "cell_type": "code", - "execution_count": 85, + "execution_count": 84, "id": "54a47e92", "metadata": {}, "outputs": [ @@ -2090,7 +2072,7 @@ "GAP: Convolution functor" ] }, - "execution_count": 85, + "execution_count": 84, "metadata": {}, "output_type": "execute_result" } @@ -2101,17 +2083,17 @@ }, { "cell_type": "code", - "execution_count": 86, + "execution_count": 85, "id": "36532210", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: E2,m1_2_2:E1->E2,m1_2_3:E1->E2,m2_3_1:E2->E3,m2_3_2:E2->E3,m2_3_3:E2->E3]\" ) ) ) / relations ) ) supported on the interval [ 0 ]>" + "GAP: " ] }, - "execution_count": 86, + "execution_count": 85, "metadata": {}, "output_type": "execute_result" } @@ -2122,7 +2104,7 @@ }, { "cell_type": "code", - "execution_count": 87, + "execution_count": 86, "id": "ce8f2fcc", "metadata": {}, "outputs": [ @@ -2144,17 +2126,17 @@ }, { "cell_type": "code", - "execution_count": 88, + "execution_count": 87, "id": "4fbf08de", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: E2,m1_2_2:E1->E2,m1_2_3:E1->E2,m2_3_1:E2->E3,m2_3_2:E2->E3,m2_3_3:E2->E3]\" ) ) ) / relations ) ) supported on the interval [ -1 .. 0 ]>" + "GAP: " ] }, - "execution_count": 88, + "execution_count": 87, "metadata": {}, "output_type": "execute_result" } @@ -2165,7 +2147,7 @@ }, { "cell_type": "code", - "execution_count": 89, + "execution_count": 88, "id": "9b1b670a", "metadata": {}, "outputs": [ @@ -2195,17 +2177,17 @@ }, { "cell_type": "code", - "execution_count": 90, + "execution_count": 89, "id": "6c0d8601", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: E2,m1_2_2:E1->E2,m1_2_3:E1->E2,m2_3_1:E2->E3,m2_3_2:E2->E3,m2_3_3:E2->E3]\" ) ) ) / relations ) ) supported on the interval [ -2 .. 0 ]>" + "GAP: " ] }, - "execution_count": 90, + "execution_count": 89, "metadata": {}, "output_type": "execute_result" } @@ -2216,7 +2198,7 @@ }, { "cell_type": "code", - "execution_count": 91, + "execution_count": 90, "id": "fdb1efec", "metadata": {}, "outputs": [ @@ -2228,9 +2210,9 @@ "\\\\\n", "\\uparrow_{\\phantom{-1}}\n", "\\\\\n", - "\\begin{pmatrix}-{m_{2,3}^{1}}&-{m_{2,3}^{2}}&-{m_{2,3}^{3}}&0&0&0\\\\ \n", - "0&-{m_{2,3}^{1}}&0&-{m_{2,3}^{2}}&-{m_{2,3}^{3}}&0\\\\ \n", - "0&0&-{m_{2,3}^{1}}&0&-{m_{2,3}^{2}}&-{m_{2,3}^{3}}\\end{pmatrix}\n", + "\\begin{pmatrix}-{m_{2,3}^{1}}&-{m_{2,3}^{2}}&0&-{m_{2,3}^{3}}&0&0\\\\ \n", + "0&-{m_{2,3}^{1}}&-{m_{2,3}^{2}}&0&-{m_{2,3}^{3}}&0\\\\ \n", + "0&0&0&-{m_{2,3}^{1}}&-{m_{2,3}^{2}}&-{m_{2,3}^{3}}\\end{pmatrix}\n", "\\\\\n", "{\\vert_{-1}}\n", "\n", @@ -2256,7 +2238,7 @@ }, { "cell_type": "code", - "execution_count": 92, + "execution_count": 91, "id": "382680b3", "metadata": {}, "outputs": [ @@ -2266,7 +2248,7 @@ "GAP: Counit ฯต : Fโˆ˜G โŸน Id of the adjunction F โŠฃ G" ] }, - "execution_count": 92, + "execution_count": 91, "metadata": {}, "output_type": "execute_result" } @@ -2277,17 +2259,17 @@ }, { "cell_type": "code", - "execution_count": 93, + "execution_count": 92, "id": "99c192c9", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: ๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) ) supported on the interval [ -2 .. 0 ]>" + "GAP: " ] }, - "execution_count": 93, + "execution_count": 92, "metadata": {}, "output_type": "execute_result" } @@ -2298,7 +2280,7 @@ }, { "cell_type": "code", - "execution_count": 94, + "execution_count": 93, "id": "4a4520db", "metadata": {}, "outputs": [ @@ -2306,32 +2288,32 @@ "data": { "text/latex": [ "$$\\begin{array}{ccc}\n", - " {\\mathcal{O}_{2}}\\oplus{\\mathcal{O}_{1}}^{\\oplus 3}\\oplus{\\mathcal{O}_{0}}^{\\oplus 6}&-\\phantom{-}{\\begin{pmatrix}{\\mathcal{O}_{2}}\\\\ \n", + " {\\mathcal{O}_{2}}\\oplus{\\mathcal{O}_{1}}^{\\oplus 3}\\oplus{\\mathcal{O}_{0}}^{\\oplus 6}&-\\phantom{-}{\\begin{pmatrix}id(\\mathcal{O}_{2})\\\\ \n", "-{y_0}\\\\ \n", "-{y_1}\\\\ \n", "-{y_2}\\\\ \n", - "{x_0y_0}\\\\ \n", - "{x_0y_1}\\\\ \n", - "{x_0y_2}\\\\ \n", - "{x_1y_1}\\\\ \n", - "{x_1y_2}\\\\ \n", - "{x_2y_2}\\end{pmatrix}}\\phantom{-}\\rightarrow&{\\mathcal{O}_{2}}\n", + "{x_0}{y_0}\\\\ \n", + "{x_1}{y_0}\\\\ \n", + "{x_1}{y_1}\\\\ \n", + "{x_2}{y_0}\\\\ \n", + "{x_2}{y_1}\\\\ \n", + "{x_2}{y_2}\\end{pmatrix}}\\phantom{-}\\rightarrow&{\\mathcal{O}_{2}}\n", " \\\\ \n", " \\uparrow_{\\phantom{-1}}&& \n", " \\uparrow_{\\phantom{-1}}\n", " \\\\ \n", - " \\begin{pmatrix}{y_0}&{\\mathcal{O}_{1}}&0&0&0&0&0&0&0&0\\\\ \n", - "{y_1}&0&{\\mathcal{O}_{1}}&0&0&0&0&0&0&0\\\\ \n", - "{y_2}&0&0&{\\mathcal{O}_{1}}&0&0&0&0&0&0\\\\ \n", - "0&-{x_0}&0&0&-{\\mathcal{O}_{0}}&0&0&0&0&0\\\\ \n", - "0&-{x_1}&0&0&0&-{\\mathcal{O}_{0}}&0&0&0&0\\\\ \n", - "0&-{x_2}&0&0&0&0&-{\\mathcal{O}_{0}}&0&0&0\\\\ \n", - "0&0&-{x_0}&0&0&-{\\mathcal{O}_{0}}&0&0&0&0\\\\ \n", - "0&0&-{x_1}&0&0&0&0&-{\\mathcal{O}_{0}}&0&0\\\\ \n", - "0&0&-{x_2}&0&0&0&0&0&-{\\mathcal{O}_{0}}&0\\\\ \n", - "0&0&0&-{x_0}&0&0&-{\\mathcal{O}_{0}}&0&0&0\\\\ \n", - "0&0&0&-{x_1}&0&0&0&0&-{\\mathcal{O}_{0}}&0\\\\ \n", - "0&0&0&-{x_2}&0&0&0&0&0&-{\\mathcal{O}_{0}}\\end{pmatrix}&&\\begin{pmatrix}\\\\\\end{pmatrix}\n", + " \\begin{pmatrix}{y_0}&id(\\mathcal{O}_{1})&0&0&0&0&0&0&0&0\\\\ \n", + "{y_1}&0&id(\\mathcal{O}_{1})&0&0&0&0&0&0&0\\\\ \n", + "{y_2}&0&0&id(\\mathcal{O}_{1})&0&0&0&0&0&0\\\\ \n", + "0&-{x_0}&0&0&-id(\\mathcal{O}_{0})&0&0&0&0&0\\\\ \n", + "0&-{x_1}&0&0&0&-id(\\mathcal{O}_{0})&0&0&0&0\\\\ \n", + "0&-{x_2}&0&0&0&0&0&-id(\\mathcal{O}_{0})&0&0\\\\ \n", + "0&0&-{x_0}&0&0&-id(\\mathcal{O}_{0})&0&0&0&0\\\\ \n", + "0&0&-{x_1}&0&0&0&-id(\\mathcal{O}_{0})&0&0&0\\\\ \n", + "0&0&-{x_2}&0&0&0&0&0&-id(\\mathcal{O}_{0})&0\\\\ \n", + "0&0&0&-{x_0}&0&0&0&-id(\\mathcal{O}_{0})&0&0\\\\ \n", + "0&0&0&-{x_1}&0&0&0&0&-id(\\mathcal{O}_{0})&0\\\\ \n", + "0&0&0&-{x_2}&0&0&0&0&0&-id(\\mathcal{O}_{0})\\end{pmatrix}&&\\begin{pmatrix}\\\\\\end{pmatrix}\n", " \\\\ \n", " \\vert_{-1} &&\\vert_{-1} \n", " \\\\ \n", @@ -2340,9 +2322,9 @@ " \\uparrow_{\\phantom{-2}}&& \n", " \\uparrow_{\\phantom{-2}}\n", " \\\\ \n", - " \\begin{pmatrix}{x_1}&-{x_0}&0&0&{\\mathcal{O}_{0}}&0&-{\\mathcal{O}_{0}}&0&0&0&0&0\\\\ \n", - "{x_2}&0&-{x_0}&0&0&{\\mathcal{O}_{0}}&0&0&0&-{\\mathcal{O}_{0}}&0&0\\\\ \n", - "0&{x_2}&-{x_1}&0&0&0&0&0&{\\mathcal{O}_{0}}&0&-{\\mathcal{O}_{0}}&0\\end{pmatrix}&&\\begin{pmatrix}\\\\\\end{pmatrix}\n", + " \\begin{pmatrix}{x_1}&-{x_0}&0&0&id(\\mathcal{O}_{0})&0&-id(\\mathcal{O}_{0})&0&0&0&0&0\\\\ \n", + "{x_2}&0&-{x_0}&0&0&id(\\mathcal{O}_{0})&0&0&0&-id(\\mathcal{O}_{0})&0&0\\\\ \n", + "0&{x_2}&-{x_1}&0&0&0&0&0&id(\\mathcal{O}_{0})&0&-id(\\mathcal{O}_{0})&0\\end{pmatrix}&&\\begin{pmatrix}\\\\\\end{pmatrix}\n", " \\\\ \n", " \\vert_{-2} &&\\vert_{-2} \n", " \\\\ \n", @@ -2361,7 +2343,7 @@ }, { "cell_type": "code", - "execution_count": 95, + "execution_count": 94, "id": "32fcae34", "metadata": {}, "outputs": [ @@ -2371,7 +2353,7 @@ "true" ] }, - "execution_count": 95, + "execution_count": 94, "metadata": {}, "output_type": "execute_result" } @@ -2382,17 +2364,17 @@ }, { "cell_type": "code", - "execution_count": 96, + "execution_count": 95, "id": "e2274a65", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: ๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations ) ) supported on the interval [ -2 .. 0 ]>" + "GAP: " ] }, - "execution_count": 96, + "execution_count": 95, "metadata": {}, "output_type": "execute_result" } @@ -2403,7 +2385,7 @@ }, { "cell_type": "code", - "execution_count": 97, + "execution_count": 96, "id": "8a376dbe", "metadata": {}, "outputs": [ @@ -2411,23 +2393,23 @@ "data": { "text/latex": [ "$$\\begin{array}{ccc}\n", - " {\\mathcal{O}_{2}}&-\\phantom{-}{\\begin{pmatrix}{\\mathcal{O}_{2}}&0&0&0&0&0&0&0&0&0\\end{pmatrix}}\\phantom{-}\\rightarrow&{\\mathcal{O}_{2}}\\oplus{\\mathcal{O}_{1}}^{\\oplus 3}\\oplus{\\mathcal{O}_{0}}^{\\oplus 6}\n", + " {\\mathcal{O}_{2}}&-\\phantom{-}{\\begin{pmatrix}id(\\mathcal{O}_{2})&0&0&0&0&0&0&0&0&0\\end{pmatrix}}\\phantom{-}\\rightarrow&{\\mathcal{O}_{2}}\\oplus{\\mathcal{O}_{1}}^{\\oplus 3}\\oplus{\\mathcal{O}_{0}}^{\\oplus 6}\n", " \\\\ \n", " \\uparrow_{\\phantom{-1}}&& \n", " \\uparrow_{\\phantom{-1}}\n", " \\\\ \n", - " \\begin{pmatrix}\\\\\\end{pmatrix}&&\\begin{pmatrix}{y_0}&{\\mathcal{O}_{1}}&0&0&0&0&0&0&0&0\\\\ \n", - "{y_1}&0&{\\mathcal{O}_{1}}&0&0&0&0&0&0&0\\\\ \n", - "{y_2}&0&0&{\\mathcal{O}_{1}}&0&0&0&0&0&0\\\\ \n", - "0&-{x_0}&0&0&-{\\mathcal{O}_{0}}&0&0&0&0&0\\\\ \n", - "0&-{x_1}&0&0&0&-{\\mathcal{O}_{0}}&0&0&0&0\\\\ \n", - "0&-{x_2}&0&0&0&0&-{\\mathcal{O}_{0}}&0&0&0\\\\ \n", - "0&0&-{x_0}&0&0&-{\\mathcal{O}_{0}}&0&0&0&0\\\\ \n", - "0&0&-{x_1}&0&0&0&0&-{\\mathcal{O}_{0}}&0&0\\\\ \n", - "0&0&-{x_2}&0&0&0&0&0&-{\\mathcal{O}_{0}}&0\\\\ \n", - "0&0&0&-{x_0}&0&0&-{\\mathcal{O}_{0}}&0&0&0\\\\ \n", - "0&0&0&-{x_1}&0&0&0&0&-{\\mathcal{O}_{0}}&0\\\\ \n", - "0&0&0&-{x_2}&0&0&0&0&0&-{\\mathcal{O}_{0}}\\end{pmatrix}\n", + " \\begin{pmatrix}\\\\\\end{pmatrix}&&\\begin{pmatrix}{y_0}&id(\\mathcal{O}_{1})&0&0&0&0&0&0&0&0\\\\ \n", + "{y_1}&0&id(\\mathcal{O}_{1})&0&0&0&0&0&0&0\\\\ \n", + "{y_2}&0&0&id(\\mathcal{O}_{1})&0&0&0&0&0&0\\\\ \n", + "0&-{x_0}&0&0&-id(\\mathcal{O}_{0})&0&0&0&0&0\\\\ \n", + "0&-{x_1}&0&0&0&-id(\\mathcal{O}_{0})&0&0&0&0\\\\ \n", + "0&-{x_2}&0&0&0&0&0&-id(\\mathcal{O}_{0})&0&0\\\\ \n", + "0&0&-{x_0}&0&0&-id(\\mathcal{O}_{0})&0&0&0&0\\\\ \n", + "0&0&-{x_1}&0&0&0&-id(\\mathcal{O}_{0})&0&0&0\\\\ \n", + "0&0&-{x_2}&0&0&0&0&0&-id(\\mathcal{O}_{0})&0\\\\ \n", + "0&0&0&-{x_0}&0&0&0&-id(\\mathcal{O}_{0})&0&0\\\\ \n", + "0&0&0&-{x_1}&0&0&0&0&-id(\\mathcal{O}_{0})&0\\\\ \n", + "0&0&0&-{x_2}&0&0&0&0&0&-id(\\mathcal{O}_{0})\\end{pmatrix}\n", " \\\\ \n", " \\vert_{-1} &&\\vert_{-1} \n", " \\\\ \n", @@ -2436,9 +2418,9 @@ " \\uparrow_{\\phantom{-2}}&& \n", " \\uparrow_{\\phantom{-2}}\n", " \\\\ \n", - " \\begin{pmatrix}\\\\\\end{pmatrix}&&\\begin{pmatrix}{x_1}&-{x_0}&0&0&{\\mathcal{O}_{0}}&0&-{\\mathcal{O}_{0}}&0&0&0&0&0\\\\ \n", - "{x_2}&0&-{x_0}&0&0&{\\mathcal{O}_{0}}&0&0&0&-{\\mathcal{O}_{0}}&0&0\\\\ \n", - "0&{x_2}&-{x_1}&0&0&0&0&0&{\\mathcal{O}_{0}}&0&-{\\mathcal{O}_{0}}&0\\end{pmatrix}\n", + " \\begin{pmatrix}\\\\\\end{pmatrix}&&\\begin{pmatrix}{x_1}&-{x_0}&0&0&id(\\mathcal{O}_{0})&0&-id(\\mathcal{O}_{0})&0&0&0&0&0\\\\ \n", + "{x_2}&0&-{x_0}&0&0&id(\\mathcal{O}_{0})&0&0&0&-id(\\mathcal{O}_{0})&0&0\\\\ \n", + "0&{x_2}&-{x_1}&0&0&0&0&0&id(\\mathcal{O}_{0})&0&-id(\\mathcal{O}_{0})&0\\end{pmatrix}\n", " \\\\ \n", " \\vert_{-2} &&\\vert_{-2} \n", " \\\\ \n", @@ -2468,17 +2450,17 @@ }, { "cell_type": "code", - "execution_count": 98, + "execution_count": 97, "id": "86c30b09", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: ๐“žโ‚,xโ‚:๐“žโ‚€->๐“žโ‚,xโ‚‚:๐“žโ‚€->๐“žโ‚,yโ‚€:๐“žโ‚->๐“žโ‚‚,yโ‚:๐“žโ‚->๐“žโ‚‚,yโ‚‚:๐“žโ‚->๐“žโ‚‚]\" ) ) ) / relations, Rows( Q ) ) )>" + "GAP: " ] }, - "execution_count": 98, + "execution_count": 97, "metadata": {}, "output_type": "execute_result" } @@ -2489,7 +2471,7 @@ }, { "cell_type": "code", - "execution_count": 99, + "execution_count": 98, "id": "46f11d58", "metadata": {}, "outputs": [ @@ -2499,7 +2481,7 @@ "true" ] }, - "execution_count": 99, + "execution_count": 98, "metadata": {}, "output_type": "execute_result" } @@ -2518,7 +2500,7 @@ }, { "cell_type": "code", - "execution_count": 100, + "execution_count": 99, "id": "d405c2f7", "metadata": {}, "outputs": [ @@ -2528,7 +2510,7 @@ "true" ] }, - "execution_count": 100, + "execution_count": 99, "metadata": {}, "output_type": "execute_result" } @@ -2555,17 +2537,17 @@ }, { "cell_type": "code", - "execution_count": 101, + "execution_count": 100, "id": "1d2b5cfa", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3)[m1_2_1:E1->E2,m1_2_2:E1->E2,m1_2_3:E1->E2,m2_3_1:E2->E3,m2_3_2:E2->E3,m2_3_3:E2->E3]\" ) ) ) / relations, Rows( Q ) )" + "GAP: PreSheaves( Q-algebroid( {E1,E2,E3}[m1_2_1:E1-โ‰ปE2,m1_2_2:E1-โ‰ปE2,m1_2_3:E1-โ‰ปE2,m2_3_1:E2-โ‰ปE3,m2_3_2:E2-โ‰ปE3,m2_3_3:E2-โ‰ปE3] ) defined by 3 objects and 6 generating morphisms, Rows( Q ) )" ] }, - "execution_count": 101, + "execution_count": 100, "metadata": {}, "output_type": "execute_result" } @@ -2576,17 +2558,17 @@ }, { "cell_type": "code", - "execution_count": 102, + "execution_count": 101, "id": "7dc0b7aa", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3)[m1_2_1:E1->E2,m1_2_2:E1->E2,m1_2_3:E1->E2,m2_3_1:E2->E3,m2_3_2:E2->E3,m2_3_3:E2->E3]\" ) ) ) / relations )" + "GAP: AdditiveClosure( Q-algebroid( {E1,E2,E3}[m1_2_1:E1-โ‰ปE2,m1_2_2:E1-โ‰ปE2,m1_2_3:E1-โ‰ปE2,m2_3_1:E2-โ‰ปE3,m2_3_2:E2-โ‰ปE3,m2_3_3:E2-โ‰ปE3] ) defined by 3 objects and 6 generating morphisms )" ] }, - "execution_count": 102, + "execution_count": 101, "metadata": {}, "output_type": "execute_result" } @@ -2597,17 +2579,17 @@ }, { "cell_type": "code", - "execution_count": 103, + "execution_count": 102, "id": "21f0d9a7", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: Homotopy category by cochains( AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3)[m1_2_1:E1->E2,m1_2_2:E1->E2,m1_2_3:E1->E2,m2_3_1:E2->E3,m2_3_2:E2->E3,m2_3_3:E2->E3]\" ) ) ) / relations ) )" + "GAP: Homotopy category by cochains( AdditiveClosure( Q-algebroid( {E1,E2,E3}[m1_2_1:E1-โ‰ปE2,m1_2_2:E1-โ‰ปE2,m1_2_3:E1-โ‰ปE2,m2_3_1:E2-โ‰ปE3,m2_3_2:E2-โ‰ปE3,m2_3_3:E2-โ‰ปE3] ) defined by 3 objects and 6 generating morphisms ) )" ] }, - "execution_count": 103, + "execution_count": 102, "metadata": {}, "output_type": "execute_result" } @@ -2618,7 +2600,7 @@ }, { "cell_type": "code", - "execution_count": 104, + "execution_count": 103, "id": "c1357390", "metadata": {}, "outputs": [], @@ -2628,7 +2610,7 @@ }, { "cell_type": "code", - "execution_count": 105, + "execution_count": 104, "id": "16c7f045", "metadata": {}, "outputs": [ @@ -2638,10 +2620,10 @@ "text": [ "Equivalence functor onto derived category of presheaves:\n", "\n", - "Homotopy category by cochains( AdditiveClosure( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3)[m1_2_1:E1->E2,m1_2_2:E1->E2,m1_2_3:E1->E2,m2_3_1:E2->E3,m2_3_2:E2->E3,m2_3_3:E2->E3]\" ) ) ) / relations ) )\n", + "Homotopy category by cochains( AdditiveClosure( Q-algebroid( {E1,E2,E3}[m1_2_1:E1-โ‰ปE2,m1_2_2:E1-โ‰ปE2,m1_2_3:E1-โ‰ปE2,m2_3_1:E2-โ‰ปE3,m2_3_2:E2-โ‰ปE3,m2_3_3:E2-โ‰ปE3] ) defined by 3 objects and 6 generating morphisms ) )\n", " |\n", " V\n", - "Derived category by cochains( PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3)[m1_2_1:E1->E2,m1_2_2:E1->E2,m1_2_3:E1->E2,m2_3_1:E2->E3,m2_3_2:E2->E3,m2_3_3:E2->E3]\" ) ) ) / relations, Rows( Q ) ) )\n" + "Derived category by cochains( PreSheaves( Q-algebroid( {E1,E2,E3}[m1_2_1:E1-โ‰ปE2,m1_2_2:E1-โ‰ปE2,m1_2_3:E1-โ‰ปE2,m2_3_1:E2-โ‰ปE3,m2_3_2:E2-โ‰ปE3,m2_3_3:E2-โ‰ปE3] ) defined by 3 objects and 6 generating morphisms, Rows( Q ) ) )\n" ] } ], @@ -2651,17 +2633,17 @@ }, { "cell_type": "code", - "execution_count": 106, + "execution_count": 105, "id": "635f2eac", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: E2,m1_2_2:E1->E2,m1_2_3:E1->E2,m2_3_1:E2->E3,m2_3_2:E2->E3,m2_3_3:E2->E3]\" ) ) ) / relations, Rows( Q ) ) ) supported on the interval [ 0 ]>" + "GAP: " ] }, - "execution_count": 106, + "execution_count": 105, "metadata": {}, "output_type": "execute_result" } @@ -2672,7 +2654,7 @@ }, { "cell_type": "code", - "execution_count": 107, + "execution_count": 106, "id": "a18b9246", "metadata": {}, "outputs": [ @@ -2682,7 +2664,7 @@ "GAP: [ 0 ]" ] }, - "execution_count": 107, + "execution_count": 106, "metadata": {}, "output_type": "execute_result" } @@ -2693,7 +2675,7 @@ }, { "cell_type": "code", - "execution_count": 108, + "execution_count": 107, "id": "95e1bdde", "metadata": {}, "outputs": [ @@ -2703,7 +2685,7 @@ "GAP: <(E1)->3, (E2)->3, (E3)->1; (m1_2_1)->3x3, (m1_2_2)->3x3, (m1_2_3)->3x3, (m2_3_1)->1x3, (m2_3_2)->1x3, (m2_3_3)->1x3>" ] }, - "execution_count": 108, + "execution_count": 107, "metadata": {}, "output_type": "execute_result" } @@ -2714,7 +2696,7 @@ }, { "cell_type": "code", - "execution_count": 109, + "execution_count": 108, "id": "7fa33c92", "metadata": {}, "outputs": [ @@ -2724,15 +2706,15 @@ "$$\\begin{array}{ccc}\n", " E_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 3} \\\\ E_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 3} \\\\ E_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 1} \\\\ \\hline & & \\\\{m_{1,2}^{1}} & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrr}\n", " \\cdot & \\cdot & \\cdot \\\\ \n", - " 1 & \\cdot & \\cdot \\\\ \n", - " \\cdot & 1 & \\cdot \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 3} \\\\ & & \\\\{m_{1,2}^{2}} & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrr}\n", " -1 & \\cdot & \\cdot \\\\ \n", + " \\cdot & -1 & \\cdot \n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 3} \\\\ & & \\\\{m_{1,2}^{2}} & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrr}\n", + " 1 & \\cdot & \\cdot \\\\ \n", " \\cdot & \\cdot & \\cdot \\\\ \n", - " \\cdot & \\cdot & 1 \n", + " \\cdot & \\cdot & -1 \n", "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 3} \\\\ & & \\\\{m_{1,2}^{3}} & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrr}\n", - " \\cdot & -1 & \\cdot \\\\ \n", - " \\cdot & \\cdot & -1 \\\\ \n", + " \\cdot & 1 & \\cdot \\\\ \n", + " \\cdot & \\cdot & 1 \\\\ \n", " \\cdot & \\cdot & \\cdot \n", "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 3} \\\\ & & \\\\{m_{2,3}^{1}} & \\mapsto & \\mathbb{Q}^{1 \\times 1}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrr}\n", " 1 & \\cdot & \\cdot \n", @@ -2753,17 +2735,17 @@ }, { "cell_type": "code", - "execution_count": 110, + "execution_count": 109, "id": "9a4a2337", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: E2,m1_2_2:E1->E2,m1_2_3:E1->E2,m2_3_1:E2->E3,m2_3_2:E2->E3,m2_3_3:E2->E3]\" ) ) ) / relations, Rows( Q ) ) ) supported on the interval [ -1 .. 0 ]>" + "GAP: " ] }, - "execution_count": 110, + "execution_count": 109, "metadata": {}, "output_type": "execute_result" } @@ -2774,7 +2756,7 @@ }, { "cell_type": "code", - "execution_count": 111, + "execution_count": 110, "id": "c6a8af8c", "metadata": {}, "outputs": [ @@ -2784,7 +2766,7 @@ "GAP: <(E1)->6, (E2)->8, (E3)->3; (m1_2_1)->8x6, (m1_2_2)->8x6, (m1_2_3)->8x6, (m2_3_1)->3x8, (m2_3_2)->3x8, (m2_3_3)->3x8>" ] }, - "execution_count": 111, + "execution_count": 110, "metadata": {}, "output_type": "execute_result" } @@ -2795,7 +2777,7 @@ }, { "cell_type": "code", - "execution_count": 112, + "execution_count": 111, "id": "78327dec", "metadata": {}, "outputs": [ @@ -2804,31 +2786,31 @@ "text/latex": [ "$$\\begin{array}{ccc}\n", " E_{1} & \\mapsto & \\mathbb{Q}^{1 \\times 6} \\\\ E_{2} & \\mapsto & \\mathbb{Q}^{1 \\times 8} \\\\ E_{3} & \\mapsto & \\mathbb{Q}^{1 \\times 3} \\\\ \\hline & & \\\\{m_{1,2}^{1}} & \\mapsto & \\mathbb{Q}^{1 \\times 8}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrrrrr}\n", - " \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & 1 \\\\ \n", - " \\cdot & \\cdot & -1 & \\cdot & \\cdot & \\cdot \\\\ \n", - " \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot \\\\ \n", - " \\cdot & \\cdot & \\cdot & \\cdot & -1 & \\cdot \\\\ \n", - " \\cdot & 1 & \\cdot & \\cdot & \\cdot & \\cdot \\\\ \n", - " \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot \\\\ \n", - " \\cdot & \\cdot & \\cdot & 1 & \\cdot & \\cdot \\\\ \n", - " \\cdot & \\cdot & \\cdot & \\cdot & 1 & \\cdot \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 6} \\\\ & & \\\\{m_{1,2}^{2}} & \\mapsto & \\mathbb{Q}^{1 \\times 8}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrrrrr}\n", + " \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & -1 \\\\ \n", + " \\cdot & \\cdot & 1 & \\cdot & \\cdot & \\cdot \\\\ \n", " \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot \\\\ \n", - " 1 & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot \\\\ \n", " \\cdot & \\cdot & \\cdot & \\cdot & 1 & \\cdot \\\\ \n", + " \\cdot & -1 & \\cdot & \\cdot & \\cdot & \\cdot \\\\ \n", " \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot \\\\ \n", - " \\cdot & \\cdot & 1 & \\cdot & \\cdot & \\cdot \\\\ \n", " \\cdot & \\cdot & \\cdot & -1 & \\cdot & \\cdot \\\\ \n", + " \\cdot & \\cdot & \\cdot & \\cdot & -1 & \\cdot \n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 6} \\\\ & & \\\\{m_{1,2}^{2}} & \\mapsto & \\mathbb{Q}^{1 \\times 8}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrrrrr}\n", " \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot \\\\ \n", - " \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & 1 \n", - "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 6} \\\\ & & \\\\{m_{1,2}^{3}} & \\mapsto & \\mathbb{Q}^{1 \\times 8}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrrrrr}\n", " -1 & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot \\\\ \n", + " \\cdot & \\cdot & \\cdot & \\cdot & -1 & \\cdot \\\\ \n", " \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot \\\\ \n", - " \\cdot & -1 & \\cdot & \\cdot & \\cdot & \\cdot \\\\ \n", " \\cdot & \\cdot & -1 & \\cdot & \\cdot & \\cdot \\\\ \n", + " \\cdot & \\cdot & \\cdot & 1 & \\cdot & \\cdot \\\\ \n", " \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot \\\\ \n", - " \\cdot & \\cdot & \\cdot & \\cdot & -1 & \\cdot \\\\ \n", - " \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & -1 \\\\ \n", + " \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & -1 \n", + "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 6} \\\\ & & \\\\{m_{1,2}^{3}} & \\mapsto & \\mathbb{Q}^{1 \\times 8}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrrrrr}\n", + " 1 & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot \\\\ \n", + " \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot \\\\ \n", + " \\cdot & 1 & \\cdot & \\cdot & \\cdot & \\cdot \\\\ \n", + " \\cdot & \\cdot & 1 & \\cdot & \\cdot & \\cdot \\\\ \n", + " \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot \\\\ \n", + " \\cdot & \\cdot & \\cdot & \\cdot & 1 & \\cdot \\\\ \n", + " \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & 1 \\\\ \n", " \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot \n", "\\end{array} \\right)}}}\\mathbb{Q}^{1 \\times 6} \\\\ & & \\\\{m_{2,3}^{1}} & \\mapsto & \\mathbb{Q}^{1 \\times 3}{\\color{blue}{\\xrightarrow{\\left( \\begin{array}{rrrrrrrr}\n", " \\cdot & \\cdot & \\cdot & -1 & \\cdot & \\cdot & \\cdot & -1 \\\\ \n", @@ -2855,17 +2837,17 @@ }, { "cell_type": "code", - "execution_count": 113, + "execution_count": 112, "id": "c66a1c4e", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: E2,m1_2_2:E1->E2,m1_2_3:E1->E2,m2_3_1:E2->E3,m2_3_2:E2->E3,m2_3_3:E2->E3]\" ) ) ) / relations, Rows( Q ) ) ) supported on the interval [ -2 .. 0 ]>" + "GAP: " ] }, - "execution_count": 113, + "execution_count": 112, "metadata": {}, "output_type": "execute_result" } @@ -2876,7 +2858,7 @@ }, { "cell_type": "code", - "execution_count": 114, + "execution_count": 113, "id": "346cdfc5", "metadata": {}, "outputs": [ @@ -2886,7 +2868,7 @@ "GAP: <(E1)->10, (E2)->15, (E3)->6; (m1_2_1)->15x10, (m1_2_2)->15x10, (m1_2_3)->15x10, (m2_3_1)->6x15, (m2_3_2)->6x15, (m2_3_3)->6x15>" ] }, - "execution_count": 114, + "execution_count": 113, "metadata": {}, "output_type": "execute_result" } @@ -2897,7 +2879,7 @@ }, { "cell_type": "code", - "execution_count": 115, + "execution_count": 114, "id": "998639c2", "metadata": {}, "outputs": [ @@ -2914,96 +2896,96 @@ "Image of <(E3)>:\n", "A row module over Q of rank 6\n", "\n", - "Image of (E1)-[{ 1*(m1_2_1) }]->(E2):\n", + "Image of <1*m1_2_1:(E1) -โ‰ป (E2)>:\n", "Source: \n", "A row module over Q of rank 15\n", "\n", "Matrix: \n", - "[ [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, -1, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, -1, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ],\n", + "[ [ 0, 0, 0, 0, -1, 0, 0, 0, 0, 0 ],\n", + " [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", " [ 0, 0, 0, 0, 0, 0, -1, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, -1, 0, 0, 0, 0 ],\n", " [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ],\n", " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, 0, 0, 0, -1, 0 ],\n", " [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],\n", + " [ 0, 0, -1, 0, 0, 0, 0, 0, 0, 0 ],\n", + " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, -1 ],\n", + " [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],\n", " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ] ]\n", + " [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ],\n", + " [ 0, 0, 0, 0, 0, -1, 0, 0, 0, 0 ],\n", + " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", + " [ 0, 0, 0, 0, 0, 0, 0, -1, 0, 0 ],\n", + " [ 0, 0, 0, 0, 0, 0, 0, 0, -1, 0 ] ]\n", "\n", "Range: \n", "A row module over Q of rank 10\n", "\n", "A morphism in Rows( Q )\n", "\n", - "Image of (E1)-[{ 1*(m1_2_2) }]->(E2):\n", + "Image of <1*m1_2_2:(E1) -โ‰ป (E2)>:\n", "Source: \n", "A row module over Q of rank 15\n", "\n", "Matrix: \n", "[ [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", + " [ -1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", + " [ 0, -1, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", + " [ 0, 0, 0, 0, 0, -1, 0, 0, 0, 0 ],\n", " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],\n", + " [ 0, 0, 0, -1, 0, 0, 0, 0, 0, 0 ],\n", " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ],\n", + " [ 0, 0, 0, 0, -1, 0, 0, 0, 0, 0 ],\n", + " [ 0, 0, 0, 0, 0, 0, 0, 0, -1, 0 ],\n", " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, 0, 0, -1, 0, 0 ],\n", + " [ 0, 0, 0, 0, 0, 0, -1, 0, 0, 0 ],\n", + " [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],\n", " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ] ]\n", + " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, -1 ] ]\n", "\n", "Range: \n", "A row module over Q of rank 10\n", "\n", "A morphism in Rows( Q )\n", "\n", - "Image of (E1)-[{ 1*(m1_2_3) }]->(E2):\n", + "Image of <1*m1_2_3:(E1) -โ‰ป (E2)>:\n", "Source: \n", "A row module over Q of rank 15\n", "\n", "Matrix: \n", - "[ [ -1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, -1, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, -1, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, -1, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, -1, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, -1, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, 0, -1, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, 0, 0, 0, -1, 0 ],\n", - " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, -1 ],\n", - " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ]\n", + "[ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", + " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", + " [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", + " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", + " [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],\n", + " [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ],\n", + " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", + " [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],\n", + " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", + " [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],\n", + " [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],\n", + " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", + " [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ],\n", + " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ],\n", + " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ]\n", "\n", "Range: \n", "A row module over Q of rank 10\n", "\n", "A morphism in Rows( Q )\n", "\n", - "Image of (E2)-[{ 1*(m2_3_1) }]->(E3):\n", + "Image of <1*m2_3_1:(E2) -โ‰ป (E3)>:\n", "Source: \n", "A row module over Q of rank 6\n", "\n", "Matrix: \n", - "[ [ 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0 \\\n", + "[ [ 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0 \\\n", "],\n", - " [ 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0 \\\n", + " [ 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0 \\\n", "],\n", - " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1 \\\n", + " [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 \\\n", "],\n", - " [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 \\\n", + " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1 \\\n", "],\n", " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 \\\n", "],\n", @@ -3015,14 +2997,14 @@ "\n", "A morphism in Rows( Q )\n", "\n", - "Image of (E2)-[{ 1*(m2_3_2) }]->(E3):\n", + "Image of <1*m2_3_2:(E2) -โ‰ป (E3)>:\n", "Source: \n", "A row module over Q of rank 6\n", "\n", "Matrix: \n", "[ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", " [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", + " [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", " [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],\n", " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],\n", " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ] ]\n", @@ -3032,14 +3014,14 @@ "\n", "A morphism in Rows( Q )\n", "\n", - "Image of (E2)-[{ 1*(m2_3_3) }]->(E3):\n", + "Image of <1*m2_3_3:(E2) -โ‰ป (E3)>:\n", "Source: \n", "A row module over Q of rank 6\n", "\n", "Matrix: \n", "[ [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", " [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", - " [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", + " [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],\n", " [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ],\n", " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],\n", " [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ] ]\n", @@ -3049,7 +3031,7 @@ "\n", "A morphism in Rows( Q )\n", "\n", - "An object in PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3)[m1_2_1:E1->E2,m1_2_2:E1->E2,m1_2_3:E1->E2,m2_3_1:E2->E3,m2_3_2:E2->E3,m2_3_3:E2->E3]\" ) ) ) / relations, Rows( Q ) ) given by the above data\n" + "An object in PreSheaves( Q-algebroid( {E1,E2,E3}[m1_2_1:E1-โ‰ปE2,m1_2_2:E1-โ‰ปE2,m1_2_3:E1-โ‰ปE2,m2_3_1:E2-โ‰ปE3,m2_3_2:E2-โ‰ปE3,m2_3_3:E2-โ‰ปE3] ) defined by 3 objects and 6 generating morphisms, Rows( Q ) ) given by the above data\n" ] } ], @@ -3077,7 +3059,7 @@ }, { "cell_type": "code", - "execution_count": 116, + "execution_count": 115, "id": "4ae76417", "metadata": {}, "outputs": [ @@ -3087,7 +3069,7 @@ "GAP: <(E1)->10, (E2)->15, (E3)->6; (m1_2_1)->15x10, (m1_2_2)->15x10, (m1_2_3)->15x10, (m2_3_1)->6x15, (m2_3_2)->6x15, (m2_3_3)->6x15>" ] }, - "execution_count": 116, + "execution_count": 115, "metadata": {}, "output_type": "execute_result" } @@ -3100,17 +3082,17 @@ }, { "cell_type": "code", - "execution_count": 117, + "execution_count": 116, "id": "73785b62", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: A strong exceptional sequence in PreSheaves( Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3)[m1_2_1:E1->E2,m1_2_2:E1->E2,m1_2_3:E1->E2,m2_3_1:E2->E3,m2_3_2:E2->E3,m2_3_3:E2->E3]\" ) ) ) / relations, Rows( Q ) )" + "GAP: A strong exceptional sequence in PreSheaves( Q-algebroid( {E1,E2,E3}[m1_2_1:E1-โ‰ปE2,m1_2_2:E1-โ‰ปE2,m1_2_3:E1-โ‰ปE2,m2_3_1:E2-โ‰ปE3,m2_3_2:E2-โ‰ปE3,m2_3_3:E2-โ‰ปE3] ) defined by 3 objects and 6 generating morphisms, Rows( Q ) )" ] }, - "execution_count": 117, + "execution_count": 116, "metadata": {}, "output_type": "execute_result" } @@ -3121,17 +3103,17 @@ }, { "cell_type": "code", - "execution_count": 118, + "execution_count": 117, "id": "40d2f308", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: Algebroid( Q, FreeCategory( RightQuiver( \"q(E1,E2,E3)[m1_2_1:E1->E2,m1_2_2:E1->E2,m1_2_3:E1->E2,m2_3_1:E2->E3,m2_3_2:E2->E3,m2_3_3:E2->E3]\" ) ) ) / relations" + "GAP: Q-algebroid( {E1,E2,E3}[m1_2_1:E1-โ‰ปE2,m1_2_2:E1-โ‰ปE2,m1_2_3:E1-โ‰ปE2,m2_3_1:E2-โ‰ปE3,m2_3_2:E2-โ‰ปE3,m2_3_3:E2-โ‰ปE3] ) defined by 3 objects and 6 generating morphisms" ] }, - "execution_count": 118, + "execution_count": 117, "metadata": {}, "output_type": "execute_result" } @@ -3142,17 +3124,17 @@ }, { "cell_type": "code", - "execution_count": 119, + "execution_count": 118, "id": "612219ab", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "GAP: q(E1,E2,E3)[m1_2_1:E1->E2,m1_2_2:E1->E2,m1_2_3:E1->E2,m2_3_1:E2->E3,m2_3_2:E2->E3,m2_3_3:E2->E3]" + "GAP: FinQuiver( \"q(E1,E2,E3)[m1_2_1:E1-โ‰ปE2,m1_2_2:E1-โ‰ปE2,m1_2_3:E1-โ‰ปE2,m2_3_1:E2-โ‰ปE3,m2_3_2:E2-โ‰ปE3,m2_3_3:E2-โ‰ปE3]\" )" ] }, - "execution_count": 119, + "execution_count": 118, "metadata": {}, "output_type": "execute_result" } @@ -3161,6 +3143,27 @@ "UnderlyingQuiver( ๐€_๐“ข )" ] }, + { + "cell_type": "code", + "execution_count": 119, + "id": "07aecaa6-2ba3-448a-9d5a-aed28d868385", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "GAP: Q-LinearClosure( PathCategory( FinQuiver( \"q(E1,E2,E3)[m1_2_1:E1-โ‰ปE2,m1_2_2:E1-โ‰ปE2,m1_2_3:E1-โ‰ปE2,m2_3_1:E2-โ‰ปE3,m2_3_2:E2-โ‰ปE3,m2_3_3:E2-โ‰ปE3]\" ) ) ) / [ -1*m1_2_1โ€ขm2_3_2 + 1*m1_2_2โ€ขm2_3_1, -1*m1_2_1โ€ขm2_3_3 + 1*m1_2_3โ€ขm2_3_1, -1*m1_2_2โ€ขm2_3_3 + 1*m1_2_3โ€ขm2_3_2 ]" + ] + }, + "execution_count": 119, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "quo_C = DefiningCategory( ๐€_๐“ข )" + ] + }, { "cell_type": "code", "execution_count": 120, @@ -3170,7 +3173,7 @@ { "data": { "text/plain": [ - "GAP: [ (E1)-[1*(m1_2_2*m2_3_1) - 1*(m1_2_1*m2_3_2)]->(E3), (E1)-[1*(m1_2_3*m2_3_1) - 1*(m1_2_1*m2_3_3)]->(E3), (E1)-[1*(m1_2_3*m2_3_2) - 1*(m1_2_2*m2_3_3)]->(E3) ]" + "GAP: [ -1*m1_2_1โ€ขm2_3_2 + 1*m1_2_2โ€ขm2_3_1:(E1) -โ‰ป (E3), -1*m1_2_1โ€ขm2_3_3 + 1*m1_2_3โ€ขm2_3_1:(E1) -โ‰ป (E3), -1*m1_2_2โ€ขm2_3_3 + 1*m1_2_3โ€ขm2_3_2:(E1) -โ‰ป (E3) ]" ] }, "execution_count": 120, @@ -3179,7 +3182,7 @@ } ], "source": [ - "RelationsOfAlgebroid( ๐€_๐“ข )" + "DefiningRelations( quo_C )" ] }, { diff --git a/DerivedCategories/gap/OnlyWithFunctorCategories.gi b/DerivedCategories/gap/OnlyWithFunctorCategories.gi index 96beaa94..11f441a4 100644 --- a/DerivedCategories/gap/OnlyWithFunctorCategories.gi +++ b/DerivedCategories/gap/OnlyWithFunctorCategories.gi @@ -4,8 +4,7 @@ # Implementations # - - +## InstallMethod( EmbeddingIntoDerivedCategory, [ IsHomotopyCategory ], @@ -18,12 +17,16 @@ InstallMethod( EmbeddingIntoDerivedCategory, add_closure := DefiningCategory( homotopy_cat ); - if not IsAlgebroid( UnderlyingCategory( add_closure ) ) then + if not IsAdditiveClosureCategory( add_closure ) then TryNextMethod( ); fi; - + B := UnderlyingCategory( add_closure ); + if not ( IsAlgebroid( B ) or IsAlgebroidFromDataTables( B ) ) then + TryNextMethod( ); + fi; + PSh := PreSheaves( B : overhead := false ); Y := IsomorphismFromSourceIntoImageOfYonedaEmbeddingOfSource( PSh ); diff --git a/DerivedCategories/tst/happel-theorem-in-copresheaves.tst b/DerivedCategories/tst/happel-theorem-in-copresheaves.tst index 447501b8..95875906 100644 --- a/DerivedCategories/tst/happel-theorem-in-copresheaves.tst +++ b/DerivedCategories/tst/happel-theorem-in-copresheaves.tst @@ -18,34 +18,25 @@ gap> f := AdditiveClosureMorphism( [ B.("vโ‚‚"), B.("vโ‚ƒ") ] / AB, [ [ B.("b") gap> U := KernelObject( I( f ) );; gap> seq := CreateStrongExceptionalSequence( [ P1, U, P2, P3 ] );; gap> seq_oid := AbstractionAlgebroid( seq );; -gap> Dimension( UnderlyingQuiverAlgebra( seq_oid ) ); -9 +gap> Assert( 0, Dimension( seq_oid ) = 9 ); gap> H := ExtendFunctorToHomotopyCategoriesByCochains( HomFunctorOfStrongExceptionalSequence( seq ) );; gap> T := ExtendFunctorToHomotopyCategoriesByCochains( TensorProductFunctorOfStrongExceptionalSequence( seq ) );; gap> epsilon := ExtendNaturalTransformationToHomotopyCategoriesByCochains( CounitOfTensorHomAdjunction( seq ) );; gap> KP4 := CreateComplex( K_coPSh_B, P4, 0 );; -gap> IsQuasiIsomorphism( PreCompose( T(QuasiIsomorphismFromProjectiveResolution( H( KP4 ), true )), epsilon( KP4 ) ) ); -true +gap> Assert( 0, IsQuasiIsomorphism( PreCompose( T(QuasiIsomorphismFromProjectiveResolution( H( KP4 ), true )), epsilon( KP4 ) ) ) ); gap> K_PSh := RangeOfFunctor( H );; gap> PSh := DefiningCategory( K_PSh );; gap> D := EquivalenceFromFullSubcategoryOfProjectivesObjectsIntoAdditiveClosureOfSource( PSh );; gap> D := ExtendFunctorToHomotopyCategoriesByCochains( D );; gap> L := LocalizationFunctorByProjectiveObjects( K_PSh );; gap> Q := D( L( H( KP4 ) ) );; -gap> IsWellDefined( Q ); -true -gap> RankOfObject( HomStructure( Q, Q ) ); -1 +gap> Assert( 0, IsWellDefined( Q ) ); +gap> Assert( 0, RankOfObject( HomStructure( Q, Q ) ) = 1 ); gap> W := CreateComplex( K_coPSh_B, DirectSum( [ P1, U, P2, P3 ] ), 0 ) / D_coPSh_B;; -gap> RankOfObject( HomStructure( W, W ) ); -9 -gap> IsZero( HomStructure( Shift( W, 1 ), W ) ) and IsZero( HomStructure( Shift( W, -1 ), W ) ); -true -gap> basis := BasisOfExternalHom( W, W );; ForAll( basis, IsWellDefined ); -true -gap> IsCongruentForMorphisms( basis[1] + basis[2] - basis[1], basis[2] ); -true -gap> HomStructure( PreCompose( [ basis[1], basis[2], basis[3] ] ) ) = PreCompose( HomStructure( basis[2] ), HomStructure( basis[1], basis[3] ) ); -true -gap> CoefficientsOfMorphism( Sum( basis ) ); -[ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] +gap> Assert( 0, RankOfObject( HomStructure( W, W ) ) = 9 ); +gap> Assert( 0, IsZero( HomStructure( Shift( W, 1 ), W ) ) and IsZero( HomStructure( Shift( W, -1 ), W ) ) ); +gap> basis := BasisOfExternalHom( W, W );; +gap> Assert( 0, ForAll( basis, IsWellDefined ) ); +gap> Assert( 0, IsCongruentForMorphisms( basis[1] + basis[2] - basis[1], basis[2] ) ); +gap> Assert( 0, HomStructure( PreCompose( [ basis[1], basis[2], basis[3] ] ) ) = PreCompose( HomStructure( basis[2] ), HomStructure( basis[1], basis[3] ) ) ); +gap> Assert( 0, CoefficientsOfMorphism( Sum( basis ) ) = [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ); diff --git a/HomotopyCategories/PackageInfo.g b/HomotopyCategories/PackageInfo.g index bddb19da..ff5c7498 100644 --- a/HomotopyCategories/PackageInfo.g +++ b/HomotopyCategories/PackageInfo.g @@ -10,7 +10,7 @@ SetPackageInfo( rec( PackageName := "HomotopyCategories", Subtitle := "Homotopy categories of additive categories", -Version := "2023.11-02", +Version := "2023.12-01", Date := (function ( ) if IsBound( GAPInfo.SystemEnvironment.GAP_PKG_RELEASE_DATE ) then return GAPInfo.SystemEnvironment.GAP_PKG_RELEASE_DATE; else return Concatenation( ~.Version{[ 1 .. 4 ]}, "-", ~.Version{[ 6, 7 ]}, "-01" ); fi; end)( ), License := "GPL-2.0-or-later", diff --git a/HomotopyCategories/examples/notebooks/strong_exceptional_sequence_in_homotopy_category_of_k_rows.ipynb b/HomotopyCategories/examples/notebooks/strong_exceptional_sequence_in_homotopy_category_of_k_rows.ipynb index ae6b3ca8..1c99c941 100644 --- a/HomotopyCategories/examples/notebooks/strong_exceptional_sequence_in_homotopy_category_of_k_rows.ipynb +++ b/HomotopyCategories/examples/notebooks/strong_exceptional_sequence_in_homotopy_category_of_k_rows.ipynb @@ -197,7 +197,7 @@ { "data": { "text/plain": [ - "GAP: Algebra( Q, FreeCategory( RightQuiver( \"q(E1)[]\" ) ) )" + "GAP: Q-algebroid( {E1}[] ) defined by 1 object and 0 generating morphisms" ] }, "execution_count": 10, @@ -245,7 +245,7 @@ "Homotopy category by cochains( Rows( Q ) )\n", " |\n", " V\n", - "Homotopy category by cochains( AdditiveClosure( Algebra( Q, FreeCategory( RightQuiver( \"q(E1)[]\" ) ) ) ) )\n" + "Homotopy category by cochains( AdditiveClosure( Q-algebroid( {E1}[] ) defined by 1 object and 0 generating morphisms ) )\n" ] } ], @@ -286,7 +286,7 @@ "text": [ "Convolution functor:\n", "\n", - "Homotopy category by cochains( AdditiveClosure( Algebra( Q, FreeCategory( RightQuiver( \"q(E1)[]\" ) ) ) ) )\n", + "Homotopy category by cochains( AdditiveClosure( Q-algebroid( {E1}[] ) defined by 1 object and 0 generating morphisms ) )\n", " |\n", " V\n", "Homotopy category by cochains( Rows( Q ) )\n" @@ -436,7 +436,7 @@ { "data": { "text/plain": [ - "GAP: " + "GAP: " ] }, "execution_count": 17, diff --git a/HomotopyCategories/gap/OnlyWithAlgebroids.gi b/HomotopyCategories/gap/OnlyWithAlgebroids.gi index 9cb84431..e7746298 100644 --- a/HomotopyCategories/gap/OnlyWithAlgebroids.gi +++ b/HomotopyCategories/gap/OnlyWithAlgebroids.gi @@ -10,7 +10,8 @@ InstallMethod( AbstractionAlgebroid, [ IsCapFullSubcategory ], function ( seq ) - local nr_vertices, arrows, q, vertices_latex, morphisms_latex, F, k, kF, o, convert_string_to_morphism, distinguished_object, range_cat, relations, oid, object_func, morphism_func, data, full_functor, f, t; + local nr_vertices, arrows, q, vertices_latex, morphisms_latex, F, k, kF, o, convert_string_to_morphism, + distinguished_object, range_cat, relations, quo_kF, oid, object_func, morphism_func, data, full_functor, f, t; nr_vertices := Length( SetOfKnownObjects( seq ) ); @@ -20,22 +21,23 @@ InstallMethod( AbstractionAlgebroid, j -> List( [ 1 .. Length( IrreducibleMorphisms( seq, [ i, j ] ) ) ], k -> [ Concatenation( "m", String(i), "_", String(j), "_", String(k) ), i, j, Concatenation( "m_{", String(i), ",", String(j), "}^{", String(k), "}" ) ] ) ) ) ) ); - q := RightQuiver( + q := FinQuiver( Concatenation( "q(", JoinStringsWithSeparator( List( [ 1 .. nr_vertices ], i -> Concatenation( "E", String( i ) ) ), "," ), - ")[", JoinStringsWithSeparator( List( arrows, m -> Concatenation( m[1], ":", String(m[2]), "->", String(m[3]) ) ), "," ), "]" ) ); + ")[", JoinStringsWithSeparator( List( arrows, m -> Concatenation( m[1], ":", "E", String(m[2]), "->", "E", String(m[3]) ) ), "," ), "]" ) ); vertices_latex := List( [ 1 .. nr_vertices ], i -> Concatenation( "E_{", String(i), "}" ) ); morphisms_latex := List( arrows, m -> m[4] ); - SetLabelsAsLaTeXStrings( q, vertices_latex, morphisms_latex ); + SetLaTeXStringsOfObjects( q, vertices_latex ); + SetLaTeXStringsOfMorphisms( q, morphisms_latex ); - F := FreeCategory( q ); + F := PathCategory( q ); range_cat := RangeCategoryOfHomomorphismStructure( seq ); - kF := Algebroid( CommutativeRingOfLinearCategory( seq ), FreeCategory( q ) : range_of_HomStructure := range_cat ); + kF := LinearClosure( CommutativeRingOfLinearCategory( seq ), F ); o := SetOfObjects( kF ); @@ -57,11 +59,13 @@ InstallMethod( AbstractionAlgebroid, List( [ 1 .. nr_vertices - 2 ], i -> Concatenation( List( [ i + 2 .. nr_vertices ], function( j ) - local H_ij, morphisms, u, coeffs, B_ij; + local H_ij, B_ij, morphisms, u, coeffs; H_ij := HomomorphismStructureOnObjects( seq, seq[i], seq[j] ); - morphisms := List( BasisPathsByVertexIndex( kF )[i][j], p -> PreComposeList( seq, List( ArrowList( p ), a -> convert_string_to_morphism( Label( a ) ) ) ) ); + B_ij := BasisOfExternalHom( kF, SetOfObjects( kF )[i], SetOfObjects( kF )[j] ); + + morphisms := List( B_ij, b -> PreComposeList( seq, seq[i], List( LabelsOfMorphisms(q){MorphismIndices( MorphismDatum(b)[2][1] )}, convert_string_to_morphism ), seq[j] ) ); morphisms := List( morphisms, m -> InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructureWithGivenObjects( seq, distinguished_object, m, H_ij ) ); @@ -69,30 +73,34 @@ InstallMethod( AbstractionAlgebroid, coeffs := EntriesOfHomalgMatrixAsListList( UnderlyingMatrix( u ) ); - B_ij := BasisOfExternalHom( kF, o[i], o[j] ); # whose elements are wrappers of the elements of BasisPathsByVertexIndex( kF )[i][j] - - return List( coeffs, coeff -> SumOfMorphisms( kF, o[i], ListN( coeff, B_ij, { c, f } -> MultiplyWithElementOfCommutativeRingForMorphisms( kF, c, f ) ), o[j] ) ); + return List( coeffs, coeff -> LinearCombinationOfMorphisms( kF, o[i], coeff, B_ij, o[j] ) ); end ) ) ) ); - oid := QuotientCategory( kF, relations : range_of_HomStructure := range_cat ); + quo_kF := QuotientCategory( kF, relations ); + + oid := AlgebroidFromDataTables( quo_kF : range_of_HomStructure := range_cat ); + + SetIsAdmissibleAlgebroid( oid, true ); + + SetDefiningCategory( oid, quo_kF ); # defining the isomorphisms from/to abstraction algebroid - object_func := o -> seq[VertexIndex( UnderlyingVertex( o ) )]; + object_func := o -> seq[ObjectIndex( o )]; morphism_func := function( s, alpha, r ) local components; - components := Paths( UnderlyingQuiverAlgebraElement( alpha ) ); + components := DecompositionIndicesOfMorphismInAlgebroid( alpha ); Assert( 0, Length( components ) <= 1 ); if IsEmpty( components ) then return ZeroMorphism( seq, s, r ); else - return PreComposeList( seq, Concatenation( [ IdentityMorphism( s ) ], List( ArrowList( components[1] ), a -> convert_string_to_morphism( Label(a) ) ) ) ); + return PreComposeList( seq, s, List( LabelsOfMorphisms( q ){components[1][2]}, convert_string_to_morphism ), r ); fi; end; @@ -108,7 +116,7 @@ InstallMethod( AbstractionAlgebroid, DeactivateCachingObject( MorphismCache( f ) ); ## - t := CapFunctor( Concatenation( "Isomorphism: strong exceptional sequence ", TEXTMTRANSLATIONS.longrightarrow, " abstraction algebroid" ), seq, oid ); + t := CapFunctor( Concatenation( "Isomorphism: strong exceptional sequence ", TEXTMTRANSLATIONS.longrightarrow, " abstraction algebroid" ), seq, oid ); AddObjectFunction( t, data[3] ); AddMorphismFunction( t, data[4] ); DeactivateCachingObject( ObjectCache( t ) ); @@ -154,7 +162,7 @@ InstallMethod( ConvolutionFunctorFromHomotopyCategoryOfAdditiveClosureOfAbstract function ( seq ) local I, F; - I := ExtendFunctorToHomotopyCategoriesByCochains( ExtendFunctorToAdditiveClosures( IsomorphismFromAbstractionAlgebroid( seq ) ) ); + I := ExtendFunctorToHomotopyCategoriesByCochains( ExtendFunctorToAdditiveClosures( IsomorphismFromAbstractionAlgebroid( seq ) ) ); F := PreCompose( I, ConvolutionFunctor( seq ) ); diff --git a/HomotopyCategories/tst/TiltingEquivalence.tst b/HomotopyCategories/tst/TiltingEquivalence.tst index fa2df53c..066f0ac4 100644 --- a/HomotopyCategories/tst/TiltingEquivalence.tst +++ b/HomotopyCategories/tst/TiltingEquivalence.tst @@ -1,14 +1,12 @@ -gap> q_O := RightQuiver( "q_O(O0,O1,O2)[x0:O0->O1,x1:O0->O1,x2:O0->O1,y0:O1->O2,y1:O1->O2,y2:O1->O2]" );; -gap> SetLabelsAsLaTeXStrings( q_O, [ "\\mathcal{O}_{0}", "\\mathcal{O}_{1}", "\\mathcal{O}_{2}" ], [ "x_0", "x_1", "x_2", "y_0", "y_1", "y_2" ] );; +gap> q_O := FinQuiver( "q_O(O0,O1,O2)[x0:O0->O1,x1:O0->O1,x2:O0->O1,y0:O1->O2,y1:O1->O2,y2:O1->O2]" );; +gap> SetLaTeXStringsOfObjects( q_O, [ "\\mathcal{O}_{0}", "\\mathcal{O}_{1}", "\\mathcal{O}_{2}" ] );; +gap> SetLaTeXStringsOfMorphisms( q_O, [ "x_0", "x_1", "x_2", "y_0", "y_1", "y_2" ] );; gap> q_O_op := OppositeQuiver( q_O );; -gap> SetLabelsAsLaTeXStrings( q_O_op, [ "\\mathcal{O}_{0}", "\\mathcal{O}_{1}", "\\mathcal{O}_{2}" ], [ "x_0", "x_1", "x_2", "y_0", "y_1", "y_2" ] );; -gap> F_O := FreeCategory( q_O );; +gap> P_O := PathCategory( q_O );; +gap> rho_O := [ [ P_O.x0y1, P_O.x1y0 ], [ P_O.x0y2, P_O.x2y0 ], [ P_O.x1y2, P_O.x2y1 ] ];; gap> QQ := HomalgFieldOfRationals( );; gap> k := QQ;; -gap> kF_O := k[F_O];; -gap> rho_O := [ PreCompose( kF_O.x0, kF_O.y1 ) - PreCompose( kF_O.x1, kF_O.y0 ), PreCompose( kF_O.x0, kF_O.y2 ) - PreCompose( kF_O.x2, kF_O.y0 ), -> PreCompose( kF_O.x1, kF_O.y2 ) - PreCompose( kF_O.x2, kF_O.y1 ) ];; -gap> A_O := AlgebroidFromDataTables( kF_O / rho_O );; +gap> A_O := AlgebroidFromDataTables( k[P_O / rho_O] );; gap> phi := 2 * A_O.x0 + 3 * A_O.x1 - A_O.x2;; gap> A_Oadd := AdditiveClosure( A_O );; gap> KA_Oadd := HomotopyCategoryByCochains( A_Oadd );; @@ -53,9 +51,7 @@ gap> Assert( 0, IsZero( HomStructure( T, Shift( T, -2 ) ) ) and gap> Assert( 0, RankOfObject( HomStructure( T, T ) ) = 12 ); gap> A_E := AbstractionAlgebroid( seq );; gap> q_E := UnderlyingQuiver( A_E );; -gap> B_E := UnderlyingQuiverAlgebra( A_E );; -gap> Assert( 0, Dimension( B_E ) = 12 ); -gap> rho_E := RelationsOfAlgebroid( A_E );; +gap> Assert( 0, Dimension( A_E ) = 12 ); gap> a := IsomorphismIntoAbstractionAlgebroid( seq );; gap> r := IsomorphismFromAbstractionAlgebroid( seq );; gap> m := A_E.("m1_2_1");;