nLab
double comma object

Double comma objects

Definition

The double comma object of three morphisms f:ADf:A\to D, g:BDg:B\to D, and h:CDh:C\to D in a 2-category can be defined as

(f/g/h)=(f/g)× B(g/h)(f/g/h) = (f/g)\times_B (g/h)

where (f/g)(f/g) and (g/h)(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

(f/g/h) [I 2,D] d 0×d 1×d 2 A×B×C f×g×h D×D×D, \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 2={abc}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 DD.

  • If A=C=1A=C=1 are the terminal category in Cat and gg is the identity functor, then f=xf=x and h=yh=y are objects of DD and (f/g/h)=(x/D/y)(f/g/h) = (x/D/y) is sometimes called the over-under-category.

  • If f,g,hf,g,h are all the identity functor of AA, then (f/g/h)(f/g/h) is the power A ()A^{(\to\to)}, the “object of composable pairs in AA.”

Revised on January 27, 2012 18:41:45 by Mike Shulman (71.136.231.206)