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Wikipedia / 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 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 limitss 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 disliked the uninformative term "comma category" (Lawvere, 1963 p.13).

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

(for source and target) are functors:

as follows:

  • .

  • respectively, such that the following diagram commutes:

.

Slice category

is the identity functor.

, and is often called the slice category or the category of making the following diagram commute:

Coslice category

The dual is an identity functor.

. It is called the coslice category, or the category of making the following diagram commute:

Arrow category

are identity functors).

.

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 is the forgetful functor mapping an abelian group to its underlying set is some fixed set (regarded as a functor from 1, which is the functor that maps a set to the free abelian group having that set as its basis. In particular, the initial object.

is called a or a . is called a or a .

, is the discrete category.

An inserter category are the two projection functors out of the product category.

Properties

For each comma category there are forgetful functors from it.

, which maps:

  • ;

  • ;

, which maps:

  • ;

  • .

, which maps:

  • ;

  • ;

Examples of use

Some notable categories

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

  • The category of pointed sets being (a functor selecting) any singleton set (the identity functor of) the category of sets. Each object of this category is a set, together with a function selecting some element of the set: the "basepoint". Morphisms are functions on sets which map basepoints to basepoints. In a similar fashion one can form the category of pointed spaces.

  • that make the diagram commute.

  • The category of graphs. In other words, the edge corresponding to a certain element of the indexing set, when translated, must be the same as the edge for the translated index.

  • . This form of comma category is often called . The nodes of the graph could be labelled in essentially the same way.

  • A category is said to be locally cartesian closed if every slice of it is cartesian closed (see above for the notion of slice). Locally cartesian closed categories are the classifying categories of dependent type theories.

Limits and universal morphisms

Limits and colimits are complete is a continuous functor produced is complete, is cocontinuous is cocomplete, and the projection functors are cocontinuous.

For example, note that in the above construction of the category of graphs as a comma category, the category of sets is complete and cocomplete, and the identity functor is continuous and 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 having a morphism to any other object in that category; it is initial. This serves to define the coproduct, when it exists.

Adjunctions

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

Natural transformations

is identical to the diagram which defines a natural transformation, 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. This is a bijective which are sections.

References

  • Lawvere, W (1963). "Functorial semantics of algebraic theories" and "Some algebraic problems in the context of functorial semantics of algebraic theories". http://www.tac.mta.ca/tac/reprints/articles/5/tr5.pdf