comma object

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.

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 $p':D\to A$, $q':D\to B$ and 2-cell $\sigma:f p'\Rightarrow g q'$ there is a 1-morphism $u:D\to(f/g)$ and isomorphisms $p u\cong p'$, $q u\cong q'$ 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 $\mu:p u \to p v$ and $\nu:q u \to q v$ 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 $p'$, $q'$, and $\sigma$ as above, there exists a *unique* $u:D\to (f/g)$ such that $p u = p'$, $q u = q'$, and $\sigma u = \alpha$. Note that any strict comma object is a comma object, but the converse is not in general true.

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

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

where $C^{\mathbf{2}}$ is the cotensor of $C$ with the arrow category $\mathbf{2} = \bullet \to \bullet$.

Suppose given a diagram

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

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.

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 $p':D\to A$, $q':D\to B$ and a natural transformation $\sigma:f p'\Rightarrow g q'$, these data specify exactly for every $d\in D$, a triple $(p'(d), q'(d), \sigma_d)$ 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.)

Revised on September 18, 2014 06:19:21
by David Roberts
(129.127.211.89)