# How to define Homology Functor in an arbitrary Abelian Category?

In the Category of Modules over a Ring, the i-th Homology of a Chain Complex is defined as the Quotient

Ker d / Im d

where d as usual denotes the differentials, indexes skipped for simplicity.

How can this be generalized to a general Abelian Category? Do I have the notion of a Quotient there?

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First an answer your question about quotients: One of the axioms of an abelian category says that every morphism has a cokernel. The quotient $B/A$ of a monomorphism $f:A \to B$ is simply its cokernel.

[Recall that the cokernel $p: B \to B/A$ is defined by the following universal property: Given a morphism $g:B \to X$ whose composition with $A \to B$ is zero there is a unique factorization $h: B/A \to X$ such that $g = hp$.]

A morphism $f: A \to B$ in an abelian category has four associated objects and five associated morphisms:

The kernel $\text{ker}\,(f): \text{Ker}\,(f) \to A$, the cokernel $\text{coker}\,(f): B \to \text{Coker}\,(f)$, the coimage $\text{Coim}\,{(f)} = \text{Coker}\,(\text{ker}\,(f))$ and the image $\text{Im}\,(f) = \text{Ker}\,(\text{coker}\,(f))$. The main axiom of abelian categories states that the canonical morphism $\hat{f}:\text{Coim}\,(f) \to \text{Im}\,(f)$ (uniquely determined by requiring that the diagram be commutative) always is an isomorphism.

Now given two morphisms $f:A' \to A$ and $g:A \to A''$ such that $gf = 0$ there are three ways to define the homology of the "complex" $A' \to A \to A''$:

1. $\text{Coker}\,(\text{Im}\,(f) \to \text{Ker}\,(g))$,
2. $\text{Ker}\,(\text{Coker}\,(f) \to \text{Coim}\,(g))$,
3. $\text{Im}\,(\text{Ker}\,(g)\to \text{Coker}\,(f))$.

The first of these corresponds to the usual $\text{Ker}/\text{Im}$ and it is not very hard to show that all three ways give canonically isomorphic objects in an abelian category. It is essential to require the category to be abelian here, the three possibilities are distinct in a general additive category (with kernels and cokernels).

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Let me just add that the possibilities 1. and 2. are dual to each other while possibility 3. is self-dual because $\text{Coim} = \text{Im}$ in an abelian category. –  t.b. Jan 19 '11 at 12:15
I am curious about how you get the diagrams to show up. Did you write it in tex somewhere else compile it, and post it as a picture? –  Sean Tilson Jan 19 '11 at 21:48

The notion of a kernel makes sense in an arbitrary abelian category (or really a category with a zero object). By definition, a kernel of a morphism $A \to B$ is an object $C$ that represents the functor $Z \to ker(\hom(Z, A) \to \hom(Z,B))$ (where the latter ker is of abelian groups). This means that homming into $Z$ is the same as homming into $A$ such that the composite to $B$ is zero.

Similarly, one can define cokernels via a universal property. The image can be defined as the kernel of the cokernel (think of what that translates to for abelian groups), or equivalently as the cokernel of the kernel (as this is one of the axioms of an abelian category). A quotient is a special case of a cokernel.

So if you have a complex in an abelian category (such that the composites of maps is zero), then one can see from the universal properties that there is a map $Im(d) \to Ker(d)$. If this is an isomorphism, then the complex is called exact. If you'd like a concrete example to try this out, you might consider for instance the category of sheaves on a topological space, for which a bunch of exercises that work things out are in chapter II of Hartshorne.

Cf. books on homological algebra and category theory, e.g. MacLane's Categories for the working mathematician.

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I think you need more than just a zero object. Another definition is as the equalizer with the zero morphism, but you need that equalizer to exist. In your definition you also need the category to be enriched over $Ab$. –  Sean Tilson Jan 19 '11 at 4:53
@Sean: Yes, you're right; the parenthetical remark would require additional explanation. (What you can do is still consider $\hom(Z,A) and$\hom(Z,B)$as pointed sets, and then define the kernel to be the object representing the kernel$\hom(Z,A) \to \hom(Z,B)\$ (where the kernel is taken for pointed sets). –  Akhil Mathew Jan 19 '11 at 12:39