The double comma object of three morphisms , , and in a 2-category can be defined as
where and are the ordinary comma objects. It can also be characterized as a 2-limit in its own right.
A double comma category is among other things the strict pullback
where is the category freely generated from a composable pair of morphisms (the linear quiver of length 2), obtained from the standard interval object in Cat by gluing it to itself. [I^{\vee 2],D]
is the functor category, i.e. the category of composable pairs of morphisms in .
If are the terminal category in Cat and is the identity functor, then and are objects of and is sometimes called the over-under-category.
If are all the identity functor of , then is the power , the “object of composable pairs in .”
Last revised on January 27, 2012 at 18:41:45. See the history of this page for a list of all contributions to it.