[Merged by Bors] - feat: computation of the connecting homomorphism of the snake lemma in concrete categories #8512
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Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
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…n concrete categories (#8512) This PR provides a lemma `ShortComplex.SnakeInput.δ_apply` which allows the computation of the connecting homomorphism in concrete categories. Incidentally, we also deduce from previous results that functors which preserve homology preserve finite limits and finite colimits. Co-authored-by: Joël Riou <37772949+joelriou@users.noreply.github.com>
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…n concrete categories (#8512) This PR provides a lemma `ShortComplex.SnakeInput.δ_apply` which allows the computation of the connecting homomorphism in concrete categories. Incidentally, we also deduce from previous results that functors which preserve homology preserve finite limits and finite colimits. Co-authored-by: Joël Riou <37772949+joelriou@users.noreply.github.com>
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… the homology sequence (#8771) This PR adds a variant of a lemma introduced in #8512: `ShortComplex.SnakeInput.δ_apply'` computes the connecting homomorphism of the snake lemma in a concrete categoriy `C` with a phrasing based on the functor `forget₂ C Ab` rather than `forget C`. From this, the lemma `ShortComplex.ShortExact.δ_apply` is obtained in a new file `Algebra.Homology.ConcreteCategory`: it gives a computation in terms of (co)cycles of the connecting homomorphism in homology attached to a short exact sequence of homological complexes in `C`. This PR also adds a lemma which computes "up to refinements" the connecting homomorphism of the homology sequence in general abelian categories.
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This PR provides a lemma
ShortComplex.SnakeInput.δ_applywhich allows the computation of the connecting homomorphism in concrete categories. Incidentally, we also deduce from previous results that functors which preserve homology preserve finite limits and finite colimits.