Comma Categories « The Unapologetic Mathematician

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 | | 3 Comments

3 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
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
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

« Previous | Next »

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.

Blogroll
Archives

May 2007

M T W T F S S

« Apr Jun »

1 2 3 4 5 6

7 8 9 10 11 12 13

14 15 16 17 18 19 20

21 22 23 24 25 26 27

28 29 30 31
Search for:
Topics

The Unapologetic Mathematician

Mathematics for the interested outsider