Comma category

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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.


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

Suppose that , , and are categories, and and (for source and target) are functors

We can form the comma category as follows:

Morphisms are composed by taking to be , whenever the latter expression is defined. The identity morphism on an object is .

Slice category

The first special case occurs when , is the identity functor, and (the category with one object and one morphism). Then for some object in . In this case, the comma category is written , and is often called the slice category over or the category of objects over . The objects can be simplified to pairs , where . Sometimes, is denoted . A morphism from to in the slice category is then an arrow making the following diagram commute:


Coslice category

The dual concept to a slice category is a coslice category. Here, has domain 1 and is an identity functor. In this case, the comma category is often written , where is the object of selected by . It is called the coslice category with respect to , or the category of objects under . The objects are pairs with . Given and , a morphism in the coslice category is a map making the following diagram commute:


Arrow category

and are identity functors on (so ). In this case, the comma category is the arrow category . Its objects are the morphisms of , and its morphisms are commuting squares in .[1]

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 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 has objects that are maps from to a set underlying a group. This relates to the left adjoint of , 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 is the canonical injection , where is the free group generated by .

An object of is called a morphism from to or a -structured arrow with domain in.[1] An object of is called a morphism from to or a -costructured arrow with codomain in.[1]

Another special case occurs when both and are functors with domain 1. If and , then the comma category , written , is the discrete category whose objects are morphisms from to .


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.

Limits and universal morphisms

Colimits in comma categories may be "inherited". If and are cocomplete, is a cocontinuous functor, and another functor (not necessarily cocontinuous), then the comma category produced will also be cocomplete{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}. For example, in the above construction of the category of graphs, the category of sets is cocomplete, and the identity functor is cocontinuous: so graphs are also cocomplete - all (small) colimits exist. This result is much harder to obtain directly.

If and are complete, and both and are continuous functors,[2] then the comma category is also complete, and the projection functors and are limit preserving.

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. For example, let be a category with the functor taking each object to and each arrow to . A universal morphism from to consists, by definition, of an object and morphism with the universal property that for any morphism there is a unique morphism with . In other words, it is an object in the comma category having a morphism to any other object in that category; it is initial. This serves to define the coproduct in , when it exists.


Lawvere showed that the functors and are adjoint if and only if the comma categories and , with and the identity functors on and respectively, are isomorphic, and equivalent elements in the comma category can be projected onto the same element of . This allows adjunctions to be described without involving sets, and was in fact the original motivation for introducing comma categories.

Natural transformations

If the domains of are equal, then the diagram which defines morphisms in with 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 Huq{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} that a natural transformation , with , corresponds to a functor which maps each object to and maps each morphism to . This is a bijective correspondence between natural transformations and functors which are sections of both forgetful functors from .


  1. 1.0 1.1 1.2 {{#invoke:citation/CS1|citation |CitationClass=book }}
  2. See I. 2.16.1 in Francis Borceux (1994), Handbook of Categorical Algebra 1, Cambridge University Press. ISBN 0-521-44178-1.

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