The double comma object of three morphisms $f:A\to D$, $g:B\to D$, and $h:C\to D$ in a 2-category can be defined as
where $(f/g)$ and $(g/h)$ 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 $I^{\vee 2} = \{a \to b \to c\}$ 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 $D$.
If $A=C=1$ are the terminal category in Cat and $g$ is the identity functor, then $f=x$ and $h=y$ are objects of $D$ and $(f/g/h) = (x/D/y)$ is sometimes called the over-under-category.
If $f,g,h$ are all the identity functor of $A$, then $(f/g/h)$ is the power $A^{(\to\to)}$, the “object of composable pairs in $A$.”
Last revised on January 27, 2012 at 18:41:45. See the history of this page for a list of all contributions to it.