preadditive categories are all Abelian

Don Aman

There are seemingly 4 different levels of linearity that a category can satisfy:

preadditivity: hom-sets are abelian groups and composition is bilinear
additivity: all finite biproducts exist
pre-Abelian: all morphisms have kernels and cokernels
Abelian: every monic is a kernel and every epic is a cokernel.

and these are all nice properties, but I wonder how useful these classifications are, since every preadditive category I've ever met is actually Abelian. OK, well I saw an additive category which wasn't Abelian in a homework exercise, but it was a rather contrived example. So does anyone have a bucket of fun examples of categories that are preadditive but not additive, additive but not pre-Abelian, or pre-Abelian but not Abelian? Do such categories occur in mathematics, or are they only for use in proving abstract nonsense theorems in category theory?

Hurkyl

Well, here are some thoughts:

It is often useful to know how things can go wrong. E.G. to know that there could be a pre-abelian category with monics that are not kernels.

Sometimes, having these classifications help you prove things. For example, the section on regular categories in Categories, Allegories (Freyd, Scedrov) has:

A is abelian iff it is an effective regular additive category.
A is abelian iff it is an exact additive category.
A is abelian iff it is an exact category with either binary products or coproducts.
A is abelian iff it is a normal category with kernels, cokernels, and either binary products or coproducts.

So, if you could manage to piece together the right things, you could then prove your category is abelian. (Not that I know what all of those terms mean! )


Another text that has been loaned to me, though I don't really understand any of it, Algebra V (Kostrikin, Shafarevich) has a whole chapter on Triangulated Categories. The first paragraph is:

1.1. Axioms. Let be an additive category. The structure of a triangulated category on is given by the following data a, b that must satisfy the axioms TR1-TR4 below.

These things are apparently supposed to be a generalization of derived categories, which are apparently related to abelian categories. Going to that chapter, we have this paragraph:

2.1. The Plan. First we introduce certain diagrams in derived categories - called distinguished triangles - that replaces and generalizes exact triples in abelian categories. The definition of such diagrams is not at all obvious. First of all, we do not even know that the category is additive: to add two morphisms we have, in a sense, to find their "common denominator". Next, although will happen to be additive, it will almost never be abelian. Therefore, we cannot apply to the standard definition of exactness.

Hurkyl

Ah, Algebra V gives examples of nonabelian additive categories that arise in nature!

It gives AbF, the category of filtered abelian groups. An object of AbF is a group X with a distinguished doubly infinite nested sequence of subgroups (that is, indexed by Z). The homomorphisms of two such objects are homomorphisms of the groups that restrict to homomorphisms of all the subgroups.

The other example is AbT of Hausdorff topological abelian groups, with morphisms the continuous group homomorphisms.

In both, it mentions how to construct a map whose kernel and cokernel are both zero, but is not an isomorphism.

(For the latter, it mentions the identity map R with discrete topology to R with the Euclidean topology)

matt grime

most interesting, in terms of current research, are categories that are definitely not abelian, but are additive with extra structure, eg, derived categories, stable categories, exact categories. these are the natural space in which to do homological algebra.

Don Aman

Apparently the category of divisable groups (abelian groups in which you can divide by integers) is not an Abelian category: Q --> Q/Z is a monic, but is not a kernel. Based on this example, I've decided that the category of fields is also not Abelian, consider for example Q-->R. And I've got your two examples, AbT and AbF. AbF is actually the one I mentioned seeing as an exercise, that I thought was rather contrived.

So I guess preadditive but not Abelian categories aren't as uncommon as I thought. I think both of those examples are actually pre-Abelian.

So now I need to convince myself that this is equivalent to the definition that I gave.

Don Aman

maybe you could give an example?

Hurkyl

The category of fields isn't even preadditive, is it? What additive structure would you put on, say, Hom(R(x), R(x))?

Don Aman

What's wrong with pointwise addition?

although maybe you're right, Hom(F,K) doesn't seem to have a zero morphism, unless you allow 0=1.

Hurkyl

The pointwise sum of two ring homomorphisms is no longer a ring homomorphism, unless the target is the zero ring!

Don Aman

Oh, of course, it won't preserve the multiplicative identity.

OK, so Fld isn't even preadditive.

matt grime

Examples? I gave you two infinite classes of interesting additive non-abelian categories: derived categories and stable categories. I dont' expect you to know what they are, they are quite specialized, and I'm not sure explaining here will work, and there is no easy example (well, that isn't trivial).

Don Aman

I looked up derived categories on wikipedia. Apparently they are categories whose objects are chain complexes. So for example does that mean that the category of chain complexes of modules is not Abelian?

Don Aman

apparently the category of vector bundles over a fixed space is additive but not pre-Abelian.

matt grime

the category of chain complexes over an abelian category with all morphisms is abelian. the category of chain complexes with morphisms that are split in each degree however is not abelian (the splitting maps need not commute with the differentials) but is exact (in particular additive).

sheaves, bundles etc are not abelian automatically. i forget which way round it is but kernels or cokernels do not exist. there is a process to make it work called sheafification. abelianicity for want of a better word fails because when you try to patch together local data to get a global object then that operation doesn't commute with kernels (let's pretend it's those for the sake of argument, but remember to go and check).

Don Aman

Can you say what an exact category is? I'm not familiar with that term.

I think neither kernels nor cokernels exist. For vector bundles anyway, I think the point is that the pointwise kernel and cokernel may not have constant dimension, and therefore they do not comprise a bundle.

I read something like this: the category of vector bundles is equivalent to the category of finitely generated locally free OX-modules, where OX is the sheaf of functions (this looks very similar to Swan's theorem). Since the category of OX-modules is abelian, we can take (co)kernels there.

I don't fully understand that statement, in part because I don't know what "locally free" means. I gather that a bundle morphism whose kernel isn't a bundle corresponds to a kernel of a module homomorphism which fails to be locally free.

Hurkyl

Freyd, Scedrov define an exact category to be one with zero, kernels, and the following property:

We already know that for x:A-->B, the middle map of
A-->Coker(Ker(x))-->Ker(Coker(x))-->B
is unique, once you've chosen your kernels and cokernels. If it's always an isomorphism, then the category is exact.

They also give the description that an exact category is precisely a category with zero in which each map A-->B factors into a kernel followed by a cokernel.


But Freyd, Scedrov also say that a category is abelian iff it is exact and additive, so I'm confused!