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Let $\mathcal{A}$ be an abelian category. I want to define the homology functor $H$ from the category $\operatorname{Ch}(\mathcal{A})$ of chain complexes in $\mathcal{A}$ to itself. The following

How to define Homology Functor in an arbitrary Abelian Category?

answers my question for objects. But what about morphisms? If $f:C\longrightarrow D.$ is a chain map, how can be defined $H(f)$?

Community
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bateman
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  • Let me link this source as a reference. https://users.math.msu.edu/users/ruiterj2/math/Documents/Notes%20and%20talks/Homology%20Functor.pdf – mticus Oct 18 '24 at 13:16

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Define $H_n(f)$ such that the composite $Ker(d^C_{n+1})\rightarrow H_n(C)\rightarrow H_{n}(D)$ coincides with $Ker(d^C_{n+1})\rightarrow Ker(d^D_{n+1})\rightarrow H_n(D)$ (first arrow induced by $f$) by using the universal properties of Ker/Coker.

Bernie
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  • How do you go from $Ker (d^C_{n+1})$ to $H_n(C)$? The only way I found is through the image of $d^C_{n+1}$, then the kernel of $d^C_n$ and then to the homology $H_n(C)$, but in this way the last two composite to zero, since they are one the cokernel of the previous – bateman Nov 12 '16 at 14:56
  • I'll link my question here since it is related : https://math.stackexchange.com/q/3745192/259363 (FFR) – Anthony Jul 04 '20 at 21:44