nLab
double comma object

Definition

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

$$(f/g/h)=(f/g){\times }_B(g/h)$$(f/g/h) = (f/g)\times_B (g/h)

where $(f/g)$ and $(g/h)$ are the ordinary comma objects. It can also be characterized as a 2-limit in its own right.

Examples

in Cat

A double comma category is among other things the strict pullback

$$\begin{array}{ccc}(f/g/h)& \to & [I^{\vee 2},D]\\ \downarrow & & {\downarrow }^{d_0\times d_1\times d_2}\\ A\times B\times C& \stackrel{f\times g\times h}{\to }& D\times D\times D\end{array},$$\array{ (f/g/h) &\to& [I^{\vee 2}, D] \\ \downarrow && \downarrow^{d_0 \times d_1 \times d_2} \\ A \times B \times C &\stackrel{f\times g \times h}{\to}& D \times D \times D } \,,

where $I^{\vee 2}=\{a\to b\to c\}$ is the category freely generated from two composable 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 $A$.”