[Merged by Bors] - feat: cochains of morphisms between cochain complexes#6701
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[Merged by Bors] - feat: cochains of morphisms between cochain complexes#6701
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jcommelin
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Thanks 🎉 Looking forward to the follow-up work.
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This PR starts the construction of the cochain complex of morphisms from a cochain complex to another.
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This looks like a spurious error bors r- |
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also, if this didn't work: |
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This PR starts the construction of the cochain complex of morphisms from a cochain complex to another.
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This PR starts the construction of the cochain complex of morphisms from a cochain complex to another.
(I am sorry this PR is a little bit longer than the norm, but I wanted to include the composition of cochains.)
If
FandGare two cochain complexes (indexed by the integers), I roughly define an element inCochain F G nto be the datum of morphismsF.X p ⟶ G.X qfor all integerspandqsuch thatp + n = q. (Before I chose this design, I also experimented with a simpler definition :F.X p ⟶ G.X (p+n)for allp, but this lead to a nightmare ofeqToHompopping everywhere.)These cochains shall play a critical role in the study of the mapping cone of a morphism of cochain complexes, and in the verification of the axioms of triangulated categories for the homotopy category, which is the reason why this PR creates the file
Algebra.Homology.HomotopyCategory.HomComplex.(Note: the composition of cochains would be an example of an instance of the type class of graded heterogeneous multiplications which I attempted to introduce in #6678. I wish that the current API for graded types could be generalized in order to involve three graded types (i.e. some
HMulrather than justMulandSMul), and that the data would involve a multiplicationX a → Y b → Z cwhenevera + b = crather than justX a → Y b → Z (a + b)., see the discussion there #6678. However, there is no hurry about a design decision here, because I shall not need more graded heterogeneous multiplications until the construction of the derived category of an abelian category and its triangulated structure is over...)