nLab comma object



2-Category theory

Limits and colimits



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.



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

(f/g) p A q α f B g C \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.


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.


(in components) More concretely:

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

In addition there is the “2-dimensional universality” saying that given u:D(f/g)u \colon D\to (f/g) and v:D(f/g)v \colon D\to (f/g) and 2-morphisms μ:pupv\mu \colon p u \to p v and ν:quqv\nu \colon q u \to q v such that αv.fμ=gν.αu\alpha v. f \mu = g\nu . \alpha u, there exists a unique 2-morphism β:uv\beta \colon u\to v such that pβ=μp\beta = \mu and qβ=νq \beta = \nu.

Notice that the 2-dimensional property implies that in the 1-dimensional property, the 1-morphism uu is unique up to unique isomorphism. A square containing a 2-cell with this property is sometimes called a comma square.


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.


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 pp', qq', and σ\sigma as above, there exists a unique u:D(f/g)u:D\to (f/g) such that pu=pp u = p', qu=qq u = q', and σu=α\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



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

f/g P A f Q C 2 dom C cod B g C \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 2C^{\mathbf{2}} is the cotensor of CC with the arrow category 2=\mathbf{2} = \bullet \to \bullet.

Pasting lemma

Suppose given a diagram

P Q A p f D h C g B \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.

  • 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 ff in f/gf/g is strong (while gg 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.

Notions of pullback:

Last revised on June 11, 2024 at 17:30:36. See the history of this page for a list of all contributions to it.