Contents

Idea

The comma category of two functors $f:C\to E$ and $g:D\to E$ is like an arrow category of $E$ where all arrows have their source in the image of $f$ and their target in the image of $g$ (and the morphisms between arrows keep track of how these sources and targets are in these images). It is a kind of 2-pullback: a directed refinement of the homotopy pullback of two functors between groupoids.

Definition

We discuss three equivalent definitions of comma categories

1. Explicitly in components

2. As a fiber product

3. As a 2-pullback

In components

As a fiber product

Let $I=\{a\to b\}$ be the (directed) interval category and $E^I=\mathrm{Funct}(I,E)$ the functor category.

The comma category is the pullback

(in the 1-category Cat of categories).

Compare this with the construction at generalized universal bundle and with the definition of loop space object.

As a 2-pullback

Alternatively, the comma category is the “lax pullback” – or rather the comma object (see the discussion at 2-limit) of the pullback diagram,

i.e. the universal cone that commutes up to a natural transformation

In terms of the imagery of loop spaces objects, the comma category is the category of directed paths in $E$ which start in the image of $f$ and end in the image of $g$.

Examples

• If $f$ and $g$ are both the identity functor of a category $C$, then $(f/g)$ is the category $C^2$ of arrows in $C$.

• If $f$ is the identity functor of $C$ and $g$ is the inclusion $1\to C$ of an object $c\in C$, then $(f/g)$ is the slice category $C/c$.

• Likewise if $g$ is the identity and $f$ is the inclusion of $c$, then $(f/g)$ is the coslice category $c/C$.

Properties

2-categorical properties

The comma category $(f/g)$ comes with a canonical 2-cell in the square

which is universal in the 2-category Cat; that is, it is an example of a 2-limit (in fact, it is a strict 2-limit). Squares with the same universal property in an arbitrary 2-category are called comma squares and their top left vertex is called a comma object.

Functors and comma categories

See at

• functors and comma categories

Further reading

a low-tech description with several special cases identified in somewhat archaic terminology