I try to define abstract homology in the following way. $\newcommand{\Ob}{\operatorname{Ob}}$ $\newcommand{\Arr}{\operatorname{Arr}}$ $\newcommand{\Com}{\operatorname{Com}}$ $\newcommand{\Fun}{\operatorname{Fun}}$ $\newcommand{\dom}{\operatorname{dom}}$ $\newcommand{\cod}{\operatorname{cod}}$ $\newcommand{\img}{\operatorname{img}}$ $\newcommand{\coker}{\operatorname{coker}}$ $\newcommand{\imgs}{\operatorname{\underline{img}}}$ $\newcommand{\kers}{\operatorname{\underline{ker}}}$ $\newcommand{\cokers}{\operatorname{\underline{coker}}}$

Let $\mathcal A$ be an abelian category. Consider the category $\Com \mathcal A$ whose objects are endomorphisms $d$ on objects $A$ with of $\mathcal A$ with $d^2 = 0$ and whose morphisms between obejcts $d$ and $d'$ of $\Com \mathcal A$ are given by arrows $f \colon A → A'$ in $\mathcal A$ (where $A = \dom d$, $A' = \dom d'$) such that $fd = d'f$.

This can be viewed as a subcategory of the arrow category $\Arr \mathcal A$.

Then kernel and image define functors $\Com \mathcal A → \Arr \mathcal A$. For any $d ∈ \Ob \Com \mathcal A$, since $d^2 = 0$, we have $\img d ≤ \ker d$, that is: there exist a uniqe morphism $η_d \colon \imgs d → \kers d$ such that $\img d = (\ker d) η_d$ (where $\imgs d = \dom (\img d)$ and $\kers d = \dom (\ker d)$).

In turns out, this defines a natural transformation $η \colon \img → \ker$. As taking domains is a functor $\Arr \mathcal A → \mathcal A$, this also defines a natural transformation $η \colon \imgs → \kers$ of functors $\Com \mathcal A → \mathcal A$.

Since $\mathcal A$ is abelian, the functor category $\Fun (\Com \mathcal A, \mathcal A)$ is as well, and we may define $$h = \cokers η = \cod (\coker η),$$ yielding a functor $$h \colon \Com \mathcal A → \mathcal A.$$

Now, I think of $\mathcal A$ as the category of $ℤ$-indexed sequences of abelian groups whose morphisms are $ℤ$-indexed sequences of homomorphisms of some degree (they may uniformly vary the index, so e.g. be homomorphisms $A_i → A_{i+1}$). I might possibly further refine $\Com \mathcal A$ to only contain endomorphisms $d$ of degree one as objects and homomorphisms $f$ of degree zero as arrows …

My questions now are

  1. Have I successfully described homology as the functor $h$ in this way?
  2. If so, is there any book that does homological algebra based on this or a similar definition in a purely categorical way?

I would especially like to see homology not defined on chain complexes, but first defined on endomorphisms $d$, and only then specified to chain complexes. (Or is this completely useless?)

  • $\begingroup$ So far I found it mentioned only briefly in Mac Lane, "Homology" and some remarks on compatibility in Spanier, "Algebraic Topology". $\endgroup$ – C-Star-W-Star Jul 21 '17 at 22:47

This is exactly how Michael Barr develops homology theory in section 2.4 of his book Acyclic Models.

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    $\begingroup$ That’s neat! Thanks for the answer. I won’t accept this, though, as Michael Barr doesn’t do homological algebra in a purely categorical way in this book (that is, without falling back to the category of modules over some ring for convenience in proving theorems). $\endgroup$ – k.stm Jul 22 '17 at 7:00
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    $\begingroup$ A reference for a purely categorical approach is Theo Buehler's survey Exact categories. You can probably combine Barr's discussion of differential objects in additive categories with Buehler's to get what you want. $\endgroup$ – Vladimir Sotirov Jul 22 '17 at 17:31

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