The comma object of two morphisms and in a 2-category is an object equipped with projections and 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 , 1-morphisms , and 2-cell there is a 1-morphism and isomorphisms , such that modulo these isomorphisms, we have . There is also an additional “2-dimensional universality” saying that given and and 2-cells and such that , there exists a unique 2-cell such that and . Note that the 2-dimensional property implies that in the 1-dimensional property, the 1-morphism 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 , , and as above, there exists a unique such that , , and . Note that any strict comma object is a comma object, but the converse is not in general true.
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 , functors , and a natural transformation , these data specify exactly for every , a triple 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 March 19, 2012 08:38:48
by Urs Schreiber