Comma category

In mathematics, a comma 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^{[citation needed]} become generally known until many years later. Several mathematical concepts can be treated as comma categories, such as the special case of a slice category. 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. The name persists even though 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 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.

Suppose that ${\displaystyle {\mathcal {A}}}$, ${\displaystyle {\mathcal {B}}}$, and ${\displaystyle {\mathcal {C}}}$ are categories, and ${\displaystyle S}$ and ${\displaystyle T}$ (for source and target) are functors:

We can form the comma category ${\displaystyle (S\downarrow T)}$ as follows:

Morphisms are composed by taking ${\displaystyle (f',g')\circ (f,g)}$ to be ${\displaystyle (f'\circ f,g'\circ g)}$, whenever the latter expression is defined. The identity morphism on an object ${\displaystyle (A,B,h)}$ is ${\displaystyle (\mathrm {id} _{A},\mathrm {id} _{B})}$.

The first special case occurs when ${\displaystyle {\mathcal {C}}={\mathcal {A}}}$, the functor ${\displaystyle S}$ is the identity functor, and ${\displaystyle {\mathcal {B}}={\textbf {1}}}$ (the category with one object ${\displaystyle *}$ and one morphism). Then ${\displaystyle T(*)=A_{*}}$ for some object ${\displaystyle A_{*}}$ in ${\displaystyle {\mathcal {A}}}$.

In this case, the comma category is written ${\displaystyle ({\mathcal {A}}\downarrow A_{*})}$, and is often called the slice category over ${\displaystyle A_{*}}$ or the category of objects over ${\displaystyle A_{*}}$. The objects ${\displaystyle (A,*,h)}$ can be simplified to pairs ${\displaystyle (A,h)}$, where ${\displaystyle h:A\rightarrow A_{*}}$. Sometimes, ${\displaystyle h}$ is denoted by ${\displaystyle \pi _{A}}$. A morphism ${\displaystyle (f,\mathrm {id} _{*})}$ from ${\displaystyle (A,\pi _{A})}$ to ${\displaystyle (A',\pi _{A'})}$ in the slice category can then be simplified to an arrow ${\displaystyle f:A\rightarrow A'}$ making the following diagram commute:

The dual concept to a slice category is a coslice category. Here, ${\displaystyle {\mathcal {C}}={\mathcal {B}}}$, ${\displaystyle S}$ has domain ${\displaystyle {\textbf {1}}}$ and ${\displaystyle T}$ is an identity functor.