My Favorite Random Fact About Abelian Categories

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  This weekend, I am off to MSRI for a two week mini-course on moduli-spaces and deformation theory.  I am very excited since it comes perfectly timed with a crescendo in my interest in the tangent space to a stack.  I can also assure all of you that this is IN NO WAY a cover for a clandestine sabotage mission aimed at the proprietors of the Secret Blogging Seminar, in a misguided attempt to foster a rivalry between the two blogs.  Absolutely none of that.

  When I get there, I will be expected to give a half hour introductory talk on one of the background concepts that we were all supposed to know.  I signed up to talk about the easiest of the availible topics, abelian categories.  This decision was in part motivated by the obscene degree to which I planned to be busy the week beforehand. However, it was also because I have a random fact about abelian categories that I enjoy sharing with otherwise knowledgable people.

  The random fact: being an abelian category is an intrinsic property of the underlying category.  This contrasts with the usual method of defining abelian categories, which first defines categories enriched over \mathbf{Ab} (called a pre-additive category by some) and then defines an abelian category as one which satisfies some axioms.  From this perspective, the abelian structure is extra data that is given.

  To see if a given (regular) category is secretly an abelian category, here’s the trick.  First, ask if the coproduct and product of any finite set of objects exists and is the same.  If so, this automatically gives the Hom sets a monoid structure as follows.  Given two arrows f,g from A to B, they factor through the map f\oplus g: A \rightarrow B\oplus B.  There is also a map id\cup id: B\cup B\rightarrow B which factors two copies of the identity map.  Since B\oplus B\sim B\cup B, you can compose f\oplus g with id\cup id; call this map f + g.  It gives every Hom set the structure of a commutative monoid (the identity is the unique map that factors through the zero object).  Next, ask if this monoid is in fact a group.  If so, the category is automatically an additive category, and all that remains is to ask if the usual abelian category axioms hold.  That is, checking whether every map has a kernel, cokernel and image. 

  Thats it!  The important thing is that every question was an arrow-theoretic question, and so was intrinsic to the category.  As a corollary, if a (regular) category can be enriched to an abelian category, this structure is unique.  The construction shows that this is really a fact about additive categories; coproducts and products coinciding is a very strong condition.

  Does anyone else have fun random facts about abelian categories I can use to spice up an otherwise by-the-numbers remedial talk?

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25 Responses to “My Favorite Random Fact About Abelian Categories”

  1. A.J. Tolland Says:

    Coincidentally, I’ll be keeping an eye on you^H^H^H^H visiting the library at MSRI next week.

  2. John Armstrong Says:

    I’m not so sure this is really “secret”. There are a lot of these sorts of enriched categories.

    Let’s say you have a category with “zero morphisms”. That is, for every pair of objects A and B there’s a morphism 0:A\rightarrow B so that 0\circ f and g\circ0 are the appropriate zero morphisms. This is “secretly” the same thing as a category enriched over \mathbf{pSet} — the category of pointed sets. We can slip back and forth between seeing it as a category that satisfies certain properties or an enriched category.

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  4. icecube Says:

    Oh that’s also my favourite fact about abelian categories! I don’t, however, know much more personally about them.

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