Comma category

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.

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).

in the slice category can then be simplified to an arrow

, a morphism in the coslice category is a map

is the forgetful functor mapping an abelian group to its underlying set, and

is some fixed set (regarded as a functor from 1), then the comma category

, is the discrete category whose objects are morphisms from

are the two projection functors out of the product category

For each comma category there are forgetful functors from it.

Limits and colimits in comma categories may be "inherited".

is another functor (not necessarily continuous), then the comma category

produced is complete,[2] and the projection functors

is cocomplete, and the projection functors are cocontinuous.

Thus, the category of graphs is complete and cocomplete.

The notion of a universal morphism to a particular colimit, or from a limit, can be expressed in terms of a comma category.

Essentially, we create a category whose objects are cones, and where the limiting cone is a terminal object; then, each universal morphism for the limit is just the morphism to the terminal object.

This works in the dual case, with a category of cocones having an initial object.

having a morphism to any other object in that category; it is initial.

William Lawvere showed that the functors

This allows adjunctions to be described without involving sets, and was in fact the original motivation for introducing comma categories.

are equal, then the diagram which defines morphisms in

is identical to the diagram which defines a natural transformation

The difference between the two notions is that a natural transformation is a particular collection of morphisms of type of the form

, while objects of the comma category contains all morphisms of type of such form.

A functor to the comma category selects that particular collection of morphisms.

This is described succinctly by an observation by S.A. Huq[3] that a natural transformation