# Comma category

In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by F. W. Lawvere (Lawvere, 1963 p.36), although the technique did not become generally known until many years later. Several mathematical concepts can be treated as comma categories. Comma categories also guarantee the existence of some limits and colimits. The name comes from the notation originally used by Lawvere, which involved the comma punctuation mark. Although standard notation has changed since the use of a comma as an operator is potentially confusing, and even Lawvere dislikes the uninformative term "comma category" (Lawvere, 1963 p.13), the name still persists.

## Definition

The most general comma category construction involves two functors with the same codomain. Often one of these will have domain 1 (the one-object one-morphism category). Some accounts of category theory consider only these special cases, but the term comma category is actually much more general.

### General form

${\mathcal {A}}{\xrightarrow {\;\;S\;\;}}{\mathcal {C}}{\xleftarrow {\;\;T\;\;}}{\mathcal {B}}$ ### Slice category ### Coslice category ### Other variations

In the case of the slice or coslice category, the identity functor may be replaced with some other functor; this yields a family of categories particularly useful in the study of adjoint functors. For example, if $T$ is the forgetful functor mapping an abelian group to its underlying set, and $s$ is some fixed set (regarded as a functor from 1), then the comma category $(s\downarrow T)$ has objects that are maps from $s$ to a set underlying a group. This relates to the left adjoint of $T$ , which is the functor that maps a set to the free abelian group having that set as its basis. In particular, the initial object of $(s\downarrow T)$ is the canonical injection $s\rightarrow T(G)$ , where $G$ is the free group generated by $s$ .

## Properties

For each comma category there are forgetful functors from it.

## Examples of use

### Some notable categories

Several interesting categories have a natural definition in terms of comma categories.