Abstract Nonsense

Crushing one theorem at a time

Abelian Categories (Pt. I)

Point of Post:  In this post we define the notion of abelian category, motivating it completely.

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We have finally come to defining the notion of abelian categories. Cool–so what are they? Well, we have slowly been building up to this point, defining \mathbf{Ab}-categories, and then preadditive categories, and then finally additive categories. So, what’s the next step? Depending on how finely we want to parse the steps we’d actually now be taking about “pre-abelian categories” (this was actually the level of fineess to which Peter Freyd cut up the definition”, but for the sake of time we’re going to concatenate the two to go straight from additive to abelian. Ok, so what exactly do we need to add to additive categories to make them abelian?

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Well, our goal the entire time we’ve been creating this progression was to create classes of categories that come closer and closer to modeling \mathbf{Ab}. So, the first thing we did was to demand that it was enriched over \mathbf{Ab} (i.e. have abelian Hom sets, with addition distributing over addition)–next we decided that we wanted zero objects, a reasonable choice–we then decided that we wanted products and coproducts to always exists, be the same, and have a characterization in terms of the additive. These are all good, they all definitely start to give us the feeling of \mathbf{Ab}–in fact, there is only one major ingredient missing. Now, I want you to take a second (don’t peek ahead!) and think what construction is ubuiquitous with our work in \mathbf{Ab} that we have yet to define? If you’re having difficulty let me give you a hint. You are on an exam, and you are given two groups (abelian if you’d like, but they don’t have to be) A,B and a group map f:A\to B. Regardless of what you are going to do with this map, or where it came from, there is always two pretty natural things to check. Ring any bells? Well, hopefully you answered kernel, and dually, cokernel. It seems that hardly one group theoretic/module theoretic argument goes by where one doesn’t use the notion of a kernel. Thus, the first thing we’d like to throw into our ever-growing list of axioms of an abelian category is that every morphism has a kernel and a cokernel.

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Ok, that surely must be it, right? What more could we possibly want out of a category? Well, unfortunately for our line of categories hopeful to get into Club Abelian, we have one more round of clip-board check-offs. Namely, we want to require something extra of our kernel and cokernel baring categories. Think about it, we definitely do use kernels and cokernels constantly in our work with abelian groups/modules–but we often use something more. Namely, if I said to you “Hey ____, here’s an epimorphism f:A\to B of abelian groups” Your first reaction is not just to examine \ker f and \text{coker} f=0. No, our well-trained (inculcated?) mathematical minds immediately jump to the theorem which has been so used, so seared into our minds that it’s a near automatic response–first isomorphism theorem. Aha! We know that X/\ker f\cong Y by a “natural” isomorphism sending x+\ker f to f(x).  It is doubtless to say that we use this first ismomorphism theorem–a lot. Thus it seems natural to require that not only do our categories have kernels and cokernels, but that their favorite associated theorem (whatever that will end up meaning in a general abelian category) should hold true.

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So, we finally get right down to brass tacks–an abelian category should be an additive one that has both kernels and cokernels and satisfies the “first isomorphism theorem”.

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Ok, besides the obvious shoring up that this definition needs in the rigor department it begs one question. We have, up until this point, convinced (hopefully) the readership that all of these restrictions on categories (requiring they be enriched over \mathbf{Ab}, have kernels and cokernels, etc.) were necessary by a pure “well, obviously” argument–if we want something to look like \mathbf{Ab} we obviously must require that it satisfies so-and-so. While this line of argument is difficult to deal with, it opens up floodgate for the eventuality that is “and that’s all the requirements we need!” I mean, if at every step along the formation of our abelian category axiom list we said “not enough!” why can we just stop now? What impartial verification do we have that this is just the right amount of requirements to say it’s “pretty close to \mathbf{Ab}.

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Well, in fact, we do have a somewhat convincing impartial argument as to why this list of axioms is enough. Namely, we have the Freyd Embedding Theorem (not going to prove that baby here) which says that every small abelian category can fully and faithfully embedded into R\text{-}\mathbf{Mod} for some ring R. While this doesn’t give us exactly what we want (it doesn’t work for the full category in general) it allows us to state that given an abelian category, the category “locally” looks like \mathbf{Ab}. So, for example, all statements involving a diagram with finitely many objects in an abelian category \mathcal{A}  can be proven just by proving it in \mathbf{Ab} since the diagram in question sits inside a small abelian subcategory of \mathcal{A}. This is precisely the kind of validation that let’s us know–yes, we have the correct definition.

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Abelian Categories

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Ok, so I’ve pretty much motivated what we’re about to talk about as much as I can, so excuse me for getting down to the nitty-gritty. In particular, we need to start by defining the objects that shall serve in our definition of “the first isomorphism theorm”.

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To begin let us define a preabelian category (yes, I know, I lied, but we shall only need this distinction for a short while) to be an additive category where every morphism has both a kernel and a cokernel. In a preabelian category there are several ways in which one may attempt to state the “first isomorphism theorem”, but really there is only one correct way. The basic idea is that we want some kind of isomorphism of the form X/\ker f\xrightarrow{\approx}\text{im }f. The only problem with this is that quotients of objects don’t make sense a priori. Thus, we have to get more clever. The key is that we need to think entirely in terms of kernels and cokernels.

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To do this we, of course, look to R\text{-}\mathbf{Mod} where we recall that if f:X\to Y is a morphism then a kernel is the inclusion \ker f\hookrightarrow X and a cokernel is the projection \pi:Y\to X/\text{im }f. So, how do we get X/\ker f entirely in terms of kernels? Well, it seems like, at least for R\text{-}\mathbf{Mod} the cokernel of the kernel, i.e. \text{coker}(\ker f\hookrightarrow X) works. Thus, this is what we do in a general preabelian. We define the coimage of the morphism X\xrightarrow{f}Y to be equal to a cokernel of the map \ker f\xrightarrow{h}X–we denote the coimage of f by \text{coim }f.

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Ok, so we have one-third of the ingredients for this first isomorphism theorem down–we have \text{coim }f which takes the place of X/\ker f. The statement of the first isomorphism theorem contains another object though, namely \text{im }f. Thus, we are left to define \text{im }f entirely in terms of kernels and cokernels. Once again, we take our cue from R\text{-}\mathbf{Mod} by noting that \text{im }f is nothing more than then kernel of the map \pi:Y\to\text{coker }f. Thus, we define the image \text{im }f to be the kernel of the map Y\to\text{coker} f.

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[1] Mac, Lane Saunders. Categories for the Working Mathematician. New York: Springer-Verlag, 1994. Print.

[2] Adámek, Jirí, Horst Herrlich, and George E. Strecker. Abstract and Concrete Categories: the Joy of Cats. New York: John Wiley & Sons, 1990. Print.

[3] Berrick, A. J., and M. E. Keating. Categories and Modules with K-theory in View. Cambridge, UK: Cambridge UP, 2000. Print.

[4] Freyd, Peter J. Abelian Categories. New York: Harper & Row, 1964. Print.

[5] Mitchell, Barry. Theory of Categories. New York: Academic, 1965. Print.

[6] Herrlich, Horst, and George E. Strecker. Category Theory: An Introduction. Lemgo: Heldermann, 2007. Print.


April 2, 2012 - Posted by | Algebra, Category Theory | , , , , , ,


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