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 $\mathcal{D}$ be an additive category. The structure of a triangulated category on $\mathcal{D}$ 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 $D(\mathcal{A})$ is additive: to add two morphisms we have, in a sense, to find their "common denominator". Next, although $D(\mathcal{A})$ will happen to be additive, it will almost never be abelian. Therefore, we cannot apply to $D(\mathcal{A})$ 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.

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

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

 

[Don Aman]

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.

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]

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

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]

What's wrong with pointwise addition?

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

 

[Don Aman]

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

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]

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).

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]

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).

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

sheaves, bundles etc are not abelian automatically. i forget which way round it is but kernels or cokernels do not exist.

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.

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).

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]

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

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!

 

[matt grime]

That isn't the exact that I meant. An exact category is an additive category eqiupped with extra structure: a class of pairs of morphisms (i,p) where i is an inflation and p a deflation. i is a kernel of p and p a cokernel of i and a load of other properties too.

Examples: Chain complexes where i and p are the short exact sequences that are split in each degree. A any additive category where i and p are split monos and split epis respectively. These are not necessarily abelian.In this language (Quillen's) an abelian category is an exact category in which every morphism factors as an inflation and a deflation.

As for the sheaves thing: the kernel or cokernel, whichever one it is is certainly a presheaf, that is on each open patch we can take (co)kernels but they do not necessarily patch together to give a global section. I think it's cokernels that don't exist. there is an example like: consider some surface, and its space locally constant functions. There is the inclusion of the globally constant functions, but the quotient by thinking modulo the constants is only a presheaf at least if the surface is not connected or something. it appears in mumford.On the larger idea of why it's hard to think of categories that are not abelian etc (unless like me you just happen to be writing a paper about exact, stable and derived categories) is that we are preprogrammed to think in terms of them. That is categories are most used exactly when we talk about abelian categories, at least to begin with. Thus categories are either very basic structures, by which I mean the canonical examples we learn when we start the subject, such as posets, groups as categories with one object and all morphisms isomorphisms, or they are abelian, ie they possesses pretty much all of the properties that one can assign to a category: kernels, pullbacks, products and their duals. Think of it like topology. We start learning topology with the basic examples of finite sets with the discrete topoology, or infinite sets with discrete, indiscrete, finite, cofinite topology, and then bang, we only do second countable hausdorff separable (essentially metric spaces over R) for ages until we actually come to more spaces we need such as spec(R). There is a reason why there is a book called counter examples in topology: the counter examples are isolated and not particulalry interesting for anything other than showing some property might fail.