Comma Categories

Another useful example of a category is a comma category. The term comes from the original notation which has since fallen out of favor because, as Saunders MacLane put it, “the comma is already overworked”.

We start with three categories $\mathcal{A}$ , $\mathcal{B}$ , and $\mathcal{C}$ , and two functors $F:\mathcal{A}\rightarrow\mathcal{C}$ and $G:\mathcal{B}\rightarrow\mathcal{C}$ . The objects of the comma category $(F\downarrow G)$ are triples $(A,x,B)$ where $A$ is an object of $\mathcal{A}$ , $B$ is an object of $\mathcal{B}$ , and $x$ is an arrow $F(A)\rightarrow G(B)$ in $\mathcal{C}$ . The morphisms are pairs $(f,g)$ — with $f$ an arrow in $\mathcal{A}$ and $g$ an arrow in $\mathcal{B}$ — making the following square commute:

$\begin{matrix}F(A_1)&\rightarrow^{F(f)}&F(A_2)\\\downarrow^{x_1}&&\downarrow^{x_2}\\G(B_1)&\rightarrow^{G(g)}&G(B_2)\end{matrix}$

So what? Well, let’s try picking $F$ to be the functor $A:\mathbf{1}\rightarrow\mathcal{A}$ sending the single object of $\mathbf{1}$ to the object $A\in\mathcal{A}$ . Then let $\mathcal{G}$ be the identity functor on $\mathcal{A}$ . Now an object of $(A\downarrow1_\mathcal{A})$ is an arrow $f:A\rightarrow B$ , where $B$ can be any other object in $\mathcal{A}$ . A morphism is then a triangle:

$\begin{matrix}A&&\\\downarrow^{f}&\searrow^{f'}&\\B&\rightarrow^{g}&B'\end{matrix}$

Work out for yourself the category $(1_\mathcal{A}\downarrow A)$ .

Here’s another example: the category $(1_\mathcal{A}\downarrow 1_\mathcal{A})$ . Verify that this is exactly the arrow category $\mathcal{A}^\mathbf{2}$ .

And another: check that given objects $A$ and $B$ in $\mathcal{A}$ , the category $(A\downarrow B)$ is the discrete category (set) $\hom_\mathcal{A}(A,B)$ .

Neat!

May 26, 2007 - Posted by John Armstrong | Category theory

11 Comments »

Oooh! But can’t you generalise? eg. Create a category (F,G,H) where objects are (A,x,B,y,C) and morphisms are triples [F(f),G(g),H(h)] and a whole host of larger counterparts? Or is it that we use cones because they are useful (eg. in limits) and such generalisations wouldn’t be.

As an aside, right at the end, when you mention (1_A,1_A), and (A,B) – aren’t those commas meant to be arrows?

Comment by PhiJ | March 24, 2008 | Reply
I’m sure you can generalize, though I don’t think I’ve seen it done, and I don’t know what they’d turn out to be useful for. There’s always plenty more to study!

And yes, those are mistakes. Thanks.

Comment by John Armstrong | March 24, 2008 | Reply
Funny – just today, in discussion with a friend, I had occasion to bring up an iterated comma (technically, an isocomma) category construction, in the context of descent theory. This particular construction is a categorification of the idea of taking a nerve of a cover.

By “nerve” of a cover p: U –> X, I mean a simplicial object where the object of 0-cells is U, the object of 1-cells is the pullback $U \times_X U$ (with its two projections to U), the object of 2-cells is the pullback $U \times_X U \times_X U$ (with its three projections to $U \times_X U$ ), and so on. So for example, 1-cells are intuitively pairs (u, u’) which become identified in X by applying p.

We may similarly consider a categorified nerve of a functor f: U –> X, where the category of 0-cells is U, the category of 1-cells is an isocomma category (f, f) [whose objects are triples (a, x, b) where a and b are objects of U and x is an isomorphism fa –> fb in X], the category of 2-cells is an isocomma category (f, f, f) along the lines PhiJ was suggesting, and so on. In the language of 2-category theory, this construction gives rise to a pseudofunctor N from the (opposite of the) simplicial category $\Delta$ to Cat, meaning that we don’t have functoriality N(f)N(g) = N(fg) on the nose, but only up to specified invertible maps (subject to compatibility conditions). Or, we can similarly define a “lax nerve”, i.e., a certain lax functor $\Delta^{op} \to Cat$ , by dropping the invertibility condition. This would be based on ordinary iterated comma categories.

And yes, this can be useful!

Comment by Todd Trimble | March 24, 2008 | Reply
What about abelian-ness of comma category? Suppose A, B are abelian categories F, G:A\rightarrow B are both right exact. Then is (F, G) abelian?

Comment by Giuss | October 27, 2009 | Reply
Giuss, that sounds like an excellent exercise. Not that I mean it sounds like homework or anything, but just that it should be instructive to try working it out, since we’ve got nice characterizations of abelian categories to work with.

Comment by John Armstrong | October 27, 2009 | Reply
- I need it to be true.
  
  Comment by Giuss | October 27, 2009 | Reply
So write it down and work it out. You need certain limits and colimits to exist, and for all monics and epics to be normal. Try writing out what these constructions “should” be (pointwise) and verify that they work.

Comment by John Armstrong | October 27, 2009 | Reply
- I can see what kernels and cokernels should be…it will work.
  
  Comment by Giuss | October 27, 2009 | Reply
The comma category (F, G) as described above is NOT abelian if F, G are both right exact. In particular it fails to have kernels. However, if G is left exact (in particular if G is identity) then it is abelian.

Comment by Giuss | November 5, 2009 | Reply
Good job.

Comment by John Armstrong | November 5, 2009 | Reply
Thanks. This makes sense, but you’re implying in comment #2 they’re useful somewhere — where?

Comment by isomorphismes | December 4, 2013 | Reply

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