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 (and, in particular, a PIE-limit).

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

Definition

Definition

(comma object)
The comma object of a pair of 1-morphisms $f \colon A\to C$ and $g \colon B\to C$ in a 2-category is an object $(f/g)$ equipped with projections $p \colon (f/g)\to A$ and $q \colon (f/g)\to B$ and a 2-morphism of this form:

$$\array{ (f/g) & \overset{p}{\longrightarrow} & A \\ \mathllap{\scriptsize{q}} \big\downarrow & \swArrow \alpha & \big\downarrow \mathrlap{\scriptsize{f}} \\ B & \underset{g}{\longrightarrow} & C }$$

which is universal in the sense of a 2-limit.

Remark

(terminology)
Comma objects are also sometimes called lax pullbacks, but this term more properly refers to the lax limit of a cospan, which is a slightly different notion.

Remark

(in components) More concretely:

Part of Def. is the statement that for any object $D$, 1-morphisms $p' \colon D\to A$, $q' \colon D\to B$ and 2-morphisms $\sigma \colon f p'\Rightarrow g q'$ there is a 1-morphism $u \colon D\to(f/g)$ and isomorphisms $p u\cong p'$, $q u\cong q'$ such that modulo these isomorphisms, we have $\sigma=\alpha u$.

In addition there is the “2-dimensional universality” saying that given $u \colon D\to (f/g)$ and $v \colon D\to (f/g)$ and 2-morphisms $\mu \colon p u \to p v$ and $\nu \colon q u \to q v$ such that $\alpha v. f \mu = g\nu . \alpha u$, there exists a unique 2-morphism $\beta \colon u\to v$ such that $p\beta = \mu$ and $q \beta = \nu$.

Notice 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.

Remark

By an iso-comma object one means the analogous notion as in Def. , now subject to the requirement that the 2-morphism is a 2-isomorphism and subject to the relevant universal property.

For more on this case see at 2-pullback.

Remark

The notion of a strict comma object is analogous to that of Def. 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.

When combining this with the constraint in Rem. , one also has the notion of strict iso-comma objects

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.

Related pages

• oplax/lax comma object

Notions of pullback:

• pullback, fiber product (limit over a cospan)

• wide pullback

• lax pullback, comma object (lax limit over a cospan)

• (∞,1)-pullback, homotopy pullback, ((∞,1)-limit over a cospan)

• base change, context extension

• pullback bundle

• contravariant functor

  • pullback in cohomology, d-invariant

  • pullback of differential forms

  • pullback of a distribution