## My Favorite Random Fact About Abelian Categories

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?

Tags: ,

### 29 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.

3. Ars Mathematica » Blog Archive » Secret of Blogging about Everything Says:

[…] both cover a broad range of advanced topics. Enjoy them while you can — as you can see here open hostilities between the two are about to break […]

4. icecube Says:

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

5. inittyorelo Says:

Hiya all, I just registered on this amazing community forum and wished to say hi! Have a memorable day!

inittyorelo

6. Cliveq Says:

Massive post websmaster!
Thats why i like this website so much!

7. vogonpomoev Says:

To all Hi! Check my Blog about technology.

8. wasaga beach Says:

wasaga beach real estate – wasaga beach real estate.

9. john1alpha Says:

http://www.sempreaggiornati.com

10. Julia Benedict Says:

It is extremely interesting for me to read the post. Thanks the author for it. I like such topics and anything that is connected to this matter. I definitely want to read more soon.

Julia Benedict
escort asian schweiz

11. marketforecaPW Says:

hello all . i learned a lot here. trading .

12. ana aslan Says:

I think you learned a lot and now you re an expert

13. Ytjvynsq Says:

pictures short scene hair,

14. smartphone Says:

You made some good points there. I did a search on the topic and found most people will agree with your blog. Enter your precise job profile in Twitter bio.

15. kredyty sigma bank Says:

If you wish for to take a great deal from this article then you have to apply
these methods to your won web site.

16. Catalina Says:

Hello mates, how is everything, and what you desire to say concerning this piece of
writing, in my view its really awesome for me.

17. Pamela Says:

Nice post. I was checking constantly this weblog and I am impressed!
Very helpful info specifically the closing phase 🙂 I care for such information
much. I used to be looking for this particular information for a
very long time. Thanks and best of luck.

18. Georgetta Says:

Always ensure that the prospective bearded dragon is active and aware of its surroundings.
It’s best if the temperature of the sand is 83f (28c). The housing should include plenty of branches as well as multiple hiding areas that act as artificial caves.

All it takes is mining tools, enough inventory space and
some free time on your hands. Spell power – increases damage and healing done
by spells. There are numerous ways to make money that include
mercenary assignments, street vending or grinding (battling in player
versus player or player versus environment mode)
are all pretty standard ways of making gold.

It is because cellulite is different from the usual body
fats you have. Walking and bike riding are great exercises to remove cellulite as they give the legs and the buttocks a great workout.
If you are considering of removing your fat deposits through the means of
a painful surgery, you need to try and reconsider your decision.

ZFFL is actually a meal planner or generator that lets you choose the food you
want to eat. Make sure the diet you choose does not
slow down your metabolism. You go further and start to
of moving, the power of smiling and other mood-improving tools.

23. http://wiki.munichjs.org/profile_domingaglcq Says:

I absolutely love your blog and find nearly all of your
post’s to be precisely what I’m looking for. Do you offer guest writers to write content for you?
I wouldn’t mind composing a post or elaborating on a number of the subjects you write with regards to here. Again, awesome site!

24. kesehatan Says:

Nice answers in return of this question with genuine arguments and explaining all on the topic of that.

25. nha pho Says:

Truly no matter if someone doesn’t know then its up to other
visitors that they will help, so here it happens.

26. Office Cleaning Says:

webpage is in fact awesome.

27. beardeddraco.com Says:

It’s awesome in favor of me to have a web page, which is helpful in favor of my knowledge.