nLab
comma object

Idea

The notion of comma object or comma square is a generalization of the notion of pullback or pullback square from category theory to 2-category theory: it is a special kind of 2-limit.

Where a pullback involves a commuting square, for a comma object this square is filled by a 2-morphism.

Definition

The comma object of two morphisms $f:A\to C$ and $g:B\to C$ in a 2-category is an object $(f/g)$ equipped with projections $p:(f/g)\to A$ and $q:(f/g)\to B$ and a 2-cell

which is universal in the sense of a 2-limit. Comma objects are also sometimes called lax pullbacks, but this term more properly refers to the lax limit of a cospan.

Part of this (to be explicit) is the statement that for any object $D$, 1-morphisms $p\prime :D\to A$, $q\prime :D\to B$ and 2-cell $\sigma :fp\prime \Rightarrow gq\prime$ there is a 1-morphism $u:D\to (f/g)$ and isomorphisms $pu\cong p\prime$, $qu\cong q\prime$ such that modulo these isomorphisms, we have $\sigma u=\alpha$. There is also an additional “2-dimensional universality” saying that given $u:D\to (f/g)$ and $v:D\to (f/g)$ and 2-cells $\mu :pu\to pv$ and $\nu :qu\to qv$ such that $\alpha v.f\mu =g\nu .\alpha u$, there exists a unique 2-cell $\beta :u\to v$ such that $p\beta =\mu$ and $q\beta =\nu$. Note that the 2-dimensional property implies that in the 1-dimensional property, the 1-morphism $u$ is unique up to unique isomorphism. A square containing a 2-cell with this property is sometimes called a comma square.

A strict comma object is analogous but has the universal property of a strict 2-limit. This means that given $p\prime$, $q\prime$, and $\sigma$ as above, there exists a unique $u:D\to (f/g)$ such that $pu=p\prime$, $qu=q\prime$, and $\sigma u=\alpha$. Note that any strict comma object is a comma object, but the converse is not in general true.

Properties

Construction

The comma object $f/g$ can be constructed by means of pullbacks and cotensors:

$$\begin{array}{ccccc}f/g& \to & P& \to & A\\ \downarrow & & \downarrow & & \downarrow f\\ Q& \to & C^2& \underset{\mathrm{dom}}{\to }& C\\ \downarrow & & \downarrow \mathrm{cod}\\ B& \underset{g}{\to }& C\end{array}$$\array{ f/g & \to & P & \to & A \\ \downarrow & & \downarrow & & \downarrow \mathrlap{\scriptsize{f}} \\ Q & \to & C^{\mathbf{2}} & \underset{dom}{\to} & C \\ \downarrow & & \downarrow \mathrlap{\scriptsize{cod}} \\ B & \underset{g}{\to} & C }

where $C^2$ is the cotensor of $C$ with the arrow category $2=•\to •$.

Pasting lemma

Suppose given a diagram

$$\begin{array}{ccccc}P& \to & Q& \to & A\\ \downarrow & & p\downarrow & ⇙& \downarrow f\\ D& \underset{h}{\to }& C& \underset{g}{\to }& B\end{array}$$\array{ P & \to & Q & \to & A \\ \downarrow & & \mathllap{\scriptsize{p}} \downarrow & \swArrow & \downarrow \mathrlap{\scriptsize{f}} \\ D & \underset{h}{\to} & C & \underset{g}{\to} & B }

where the right-hand square is a comma square. Then the following are equivalent:

• the whole diagram is a comma square
• the left-hand square is a (2-)pullback square

The proof is analogous to that at pullback.

Examples

In Cat, a comma category is a comma object (in fact a strict one, as normally defined); these give their name to the general notion.

Eduardo Pareja-Tobes?: Not sure about this but, with the strict definition I think you end up having specified isos all around at the level of morphisms; comma categories as normally defined are comma objects in Cat, but not strict ones (of course they’re equivalent to the strict ones). I remember reading something like this in Makkai-Paré Accessible categories book

Mike Shulman: As far as I can tell, they are strict. Given $D$, functors $p\prime :D\to A$, $q\prime :D\to B$ and a natural transformation $\sigma :fp\prime \Rightarrow gq\prime$, these data specify exactly for every $d\in D$, a triple $(p\prime (d),q\prime (d),{\sigma }_d)$ which is an object of the comma category. Perhaps you are remembering a related remark about pseudo-pullbacks versus iso-comma objects? (If you post your comments at the nForum, for instance on this discussion, other people will be more likely to see it.)