nLab comma object

Contents

Context

2-Category theory

2-category theory

Structures on 2-categories

Limits and colimits

limits and colimits

Contents

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 $p':D\to A$, $q':D\to B$ 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 $\sigma=\alpha u$. 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.

Properties

Construction

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 $\mathbf{2} = \bullet \to \bullet$.

Pasting lemma

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.

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.

• In the 2-category of virtual double categories, a comma object is a comma double category. If the virtual double categories are (pseudo) double categories and the domain functor $f$ in $f/g$ is strong (while $g$ might be only lax), then the comma object is also a pseudo double category and the comma object lives in the 2-category of pseudo double categories and lax functors.

Last revised on December 28, 2018 at 04:24:44. See the history of this page for a list of all contributions to it.