nLab
comma object

Context

2-Category theory

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:ACf:A\to C and g:BCg:B\to C in a 2-category is an object (f/g)(f/g) equipped with projections p:(f/g)Ap:(f/g)\to A and q:(f/g)Bq:(f/g)\to B and a 2-cell

Comma Square ( f / g ) (f/g) A A B B C C f f g g q q p p α \alpha

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 DD, 1-morphisms p:DAp':D\to A, q:DBq':D\to B and 2-cell σ:fpgq\sigma:f p'\Rightarrow g q' there is a 1-morphism u:D(f/g)u:D\to(f/g) and isomorphisms pupp u\cong p', quqq u\cong q' such that modulo these isomorphisms, we have σu=α\sigma u=\alpha. There is also an additional “2-dimensional universality” saying that given u:D(f/g)u:D\to (f/g) and v:D(f/g)v:D\to (f/g) and 2-cells μ:pupv\mu:p u \to p v and ν:quqv\nu:q u \to q v such that αv.fμ=gν.αu\alpha v. f \mu = g\nu . \alpha u, there exists a unique 2-cell β:uv\beta:u\to v such that pβ=μp\beta = \mu and qβ=νq \beta = \nu. Note 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.

A strict comma object is analogous 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.

Properties

Construction

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.

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 DD, functors p:DAp':D\to A, q:DBq':D\to B and a natural transformation σ:fpgq\sigma:f p'\Rightarrow g q', these data specify exactly for every dDd\in D, a triple (p(d),q(d),σ d)(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)