nLab
comma category

Idea

The comma category of two functors $f:C\to E$ and $g:D\to E$ is like an arrow category of $E$ where all arrows have their source in the image of $f$ and their target in the image of $g$ (and the morphisms between arrows keep track of how these sources and targets are in these images).

Definition

If $f:C\to E$ and $g:D\to E$ are functors, their comma category is the category $(f/g)$ whose

• objects are triples $(c,d,\alpha )$ where $c\in C$, $d\in D$, and $\alpha :f(c)\to g(d)$ is a morphism in $E$, and whose

• morphisms from $(c_1,d_1,{\alpha }_1)$ to $(c_2,d_2,{\alpha }_2)$ are pairs $(\beta ,\gamma )$, where $\beta :c_1\to c_2$ and $\gamma :d_1\to d_2$ are morphisms in $C$ and $D$, respectively, such that ${\alpha }_2.f(\beta )=g(\gamma ).{\alpha }_1$.

$$\begin{array}{ccc}f(c_1)& \stackrel{f(\beta )}{\to }& f(c_2)\\ {\downarrow }^{{\alpha }_1}& & {\downarrow }^{{\alpha }_2}\\ g(d_1)& \stackrel{g(\gamma )}{\to }& g(d_2)\\ \\ (c_1,d_1,{\alpha }_1)& \stackrel{(\beta ,\gamma )}{\to }& (c_2,d_2,{\alpha }_2)\end{array}$$\array{ f(c_1) &\stackrel{f(\beta)}{\to}& f(c_2) \\ \downarrow^{\alpha_1} && \downarrow^{\alpha_2} \\ g(d_1) &\stackrel{g(\gamma)}{\to}& g(d_2) \\ \\ (c_1,d_1, \alpha_1) &\stackrel{(\beta,\gamma)}{\to}& (c_2,d_2, \alpha_2) }

Another common notation for the comma category is $(f\downarrow g)$. The original notation, from which the terminology is derived, is $(f,g)$, but this is rarely used any more; see the discussion below.

Definition in terms of pullbacks

Let $I=\{a\to b\}$ be the (directed) interval category and $E^I=\mathrm{Funct}(I,E)$ the functor category.

The comma category is the pullback

$$\begin{array}{ccc}(f/g)& \to & E^I\\ \downarrow & & {\downarrow }^{d_0\times d_1}\\ C\times D& \stackrel{f\times g}{\to }& E\times E\end{array}$$\array{ (f/g) &\to& E^I \\ \downarrow && \downarrow^{d_0 \times d_1} \\ C \times D &\stackrel{f \times g}{\to}& E \times E }

(in the 1-category of categories).

Compare this with the construction at generalized universal bundle and with the definition of loop space object.

Alternatively, the comma category is the “lax pullback” – or rather the comma object (see the discussion at 2-limit) of the pullback diagram,

$$\begin{array}{ccc}& & C\\ & & {\downarrow }^f\\ D& \stackrel{g}{\to }& E\end{array}$$\array{ && C \\ && \downarrow^f \\ D &\stackrel{g}{\to}& E }

i.e. the universal cone that commutes up to a natural transformation

$$\begin{array}{ccc}(f/g)& \to & C\\ \downarrow & \Downarrow & {\downarrow }^f\\ D& \stackrel{g}{\to }& E\end{array}$$\array{ (f/g) &\to& C \\ \downarrow &\Downarrow& \downarrow^f \\ D &\stackrel{g}{\to}& E }

In terms of the imagery of loop spaces objects, the comma category is the category of directed paths in $E$ which start in the image of $f$ and end in the image of $g$.

Examples

• If $f$ and $g$ are both the identity functor of a category $C$, then $(f/g)$ is the category $C^2$ of arrows in $C$.

• If $f$ is the identity functor of $C$ and $g$ is the inclusion $1\to C$ of an object $c\in C$, then $(f/g)$ is the slice category $C/c$.

• Likewise if $g$ is the identity and $f$ is the inclusion of $c$, then $(f/g)$ is the coslice category $c/C$.

2-categorical properties

The comma category $(f/g)$ comes with a canonical 2-cell in the square

which is universal in the 2-category Cat; that is, it is an example of a 2-limit (in fact, it is a strict 2-limit). Squares with the same universal property in an arbitrary 2-category are called comma squares and their top left vertex is called a comma object.

Further properties

• functors and comma categories

Discussion about notation

The terminology “comma category” is a holdover from the original notation $(f,g)$ for such a category, which generalises $(x,y)$ or $C(x,y)$ for a hom-set.

It's a very natural notation, as it generalises the notation $(x,y)$ (or $[x,y]$ as is now more common) for a hom-set. But personally, I like $(f\to g)$ (or $(f\searrow g)$ if you want to differentiate from a cocomma category, but that seems an unlikely confusion), as it is a category of arrows from $f$ to $g$. —Toby Bartels

Mike: Perhaps. I never write $(x,y)$ for a hom-set, only $A(x,y)$ or ${\mathrm{hom}}_A(x,y)$ where $A$ is the category involved, and this is also the common practice in nearly all mathematics I have read. I have seen $[x,y]$ for an internal-hom object in a closed monoidal category, and for a hom-set in a homotopy category, but not for a hom-set in an arbitrary category.

I would be okay with calling the comma category (or more generally the comma object) $E(f,g)$ or ${\mathrm{hom}}_E(f,g)$ if you are considering it as a discrete fibration from $A$ to $B$. But if you are considering it as a category in its own right, I think that such notation is confusing. I don’t mind the arrow notations, but I prefer $(f/g)$ as less visually distracting, and evidently a generalization of the common notation $C/x$ for a slice category.

Toby: Well, I never stick ‘$E$’ in there unless necessary to avoid ambiguity. I agree that the slice-generalising notation is also good. I'll use it too, but I edited the text to not denigrate the hom-set generalising notation so much.

Mike: The main reason I don’t like unadorned $(f,g)$ for either comma objects or hom-sets is that it’s already such an overloaded notation. My first thought when I see $(f,g)$ in a category is that we have $f:X\to A$ and $g:X\to B$ and we’re talking about the pair $(f,g):X\to A\times B$ — surely also a natural generalization of the very well-established notation for ordered pairs.

Toby: The notation $(f/g/h)$ for a double comma object makes me like $(f\to g\to h)$ even more!

Mike: I’d rather avoid using $\to$ in the name of an object; talking about projections $p:(f\to g)\to A$ looks a good deal more confusing to me than $p:(f/g)\to A$.

Toby: I can handle that, but after thinking about it more, I've realised that the arrow doesn't really work. If $f,g:A\to B$, then $f\to g$ ought to be the set of transformations between them. (Or $f\Rightarrow g$, but you can't keep that decoration up.)

Mike: Let me summarize this discussion so far, and try to get some other people into it. So far the only argument I have heard in favor of the notation $(f,g)$ is that it generalizes a notation for hom-sets. In my experience that notation for hom-sets is rare-to-nonexistent, nor do I like it as a notation for hom-sets: for one thing it doesn’t indicate the category in question, and for another it looks like an ordered pair. The notation $(f,g)$ for a comma category also looks like an ordered pair, which it isn’t. I also don’t think that a comma category is very much like a hom-set; it happens to be a hom-set when the domains of $f$ and $g$ are the point, but in general it seems to me that a more natural notion of hom-set between functors is a set of natural transformations. It’s really the fibers of the comma category, considered as a fibration from $C$ to $D$, that are hom-sets. Finally, I don’t think the notation $(f,g)$ scales well to double comma objects; we could write $(f,g,h)$ but it is now even less like a hom-set.

Urs: to be frank, I used it without thinking much about it. Which of the other two is your favorite? By the way, Kashiwara-Schapira use $M[C\stackrel{f}{\to }E\stackrel{g}{\leftarrow }D]$. Maybe $\mathrm{comma}[C\stackrel{f}{\to }E\stackrel{g}{\leftarrow }D]$? Lengthy, but at least unambiguous. Or maybe ${}_f{E^I}_g$?

Zoran Skoda: $(f/g)$ or $(f\downarrow g)$ are the only two standard notations nowdays, I think the original $(f,g)$ which was done for typographical reasons in archaic period is abandonded by the LaTeX era. $(f/g)$ is more popular among practical mathematicians, and special cases, like when $g={\mathrm{id}}_D$) and $(f\downarrow g)$ among category experts…other possibilities for notation should be avoided I think.

Urs: sounds good. I’ll try to stick to $(f/g)$ then.

Mike: There are many category theorists who write $(f/g)$, including (in my experience) most Australians. I prefer $(f/g)$ myself, although I occasionally write $(f\downarrow g)$ if I’m talking to someone who I worry might be confused by $(f/g)$.

Urs: recently in a talk when an over-category appeared as $C/a$ somebody in the audience asked: “What’s that quotient?”. But $(C/a)$ already looks different. And of course the proper $({\mathrm{Id}}_C/{\mathrm{const}}_a)$ even more so.

Anyway, that just to say: i like $(f/g)$, find it less cumbersome than $(f\downarrow g)$ and apologize for having written $(f,g)$ so often.

Toby: I find $(f\downarrow g)$ more self explanatory, but $(f/g)$ is cool. $(f,g)$ was reasonable, but we now have better options.

Further reading

a low-tech description with several special cases identified in somewhat archaic terminology