Comma category

  (Redirected from Arrow 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[citation needed] 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. 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).

DefinitionEdit

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 formEdit

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

 

We can form the comma category   as follows:

  • The objects are all triples   with   an object in  ,   an object in  , and   a morphism in  .
  • The morphisms from   to   are all pairs   where   and   are morphisms in   and   respectively, such that the following diagram commutes:

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

Slice categoryEdit

The first special case occurs when  , the functor   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 by  . A morphism   from   to   in the slice category can then be simplified to an arrow   making the following diagram commute:

Coslice categoryEdit

The dual concept to a slice category is a coslice category. Here,  ,   has domain   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 categoryEdit

  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 variationsEdit

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

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

An inserter category is a (non-full) subcategory of the comma category where   and   are required. The comma category can also be seen as the inserter of   and  , where   and   are the two projection functors out of the product category  .

PropertiesEdit

For each comma category there are forgetful functors from it.

  • Domain functor,  , which maps:
    • objects:  ;
    • morphisms:  ;
  • Codomain functor,  , which maps:
    • objects:  ;
    • morphisms:  .
  • Arrow functor,  , which maps:
    • objects:  ;
    • morphisms:  ;

Examples of useEdit

Some notable categoriesEdit

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

  • The category of pointed sets is a comma category,   with   being (a functor selecting) any singleton set, and   (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  .
  • The category of associative algebras over a ring   is the coslice category  , since any ring homomorphism   induces an associative  -algebra structure on  , and vice versa. Morphisms are then maps   that make the diagram commute.
  • The category of graphs is  , with   the functor taking a set   to  . The objects   then consist of two sets and a function;   is an indexing set,   is a set of nodes, and   chooses pairs of elements of   for each input from  . That is,   picks out certain edges from the set   of possible edges. A morphism in this category is made up of two functions, one on the indexing set and one on the node set. They must "agree" according to the general definition above, meaning that   must satisfy  . 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.
  • Many "augmentation" or "labelling" operations can be expressed in terms of comma categories. Let   be the functor taking each graph to the set of its edges, and let   be (a functor selecting) some particular set: then   is the category of graphs whose edges are labelled by elements of  . This form of comma category is often called objects  -over   - closely related to the "objects over  " discussed above. Here, each object takes the form  , where   is a graph and   a function from the edges of   to  . 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 morphismsEdit

Limits and colimits in comma categories may be "inherited". If   and   are complete,   is a continuous functor, and   is another functor (not necessarily continuous), then the comma category   produced is complete,[2] and the projection functors   and   are continuous. Similarly, if   and   are cocomplete, and   is cocontinuous, then   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; 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.

AdjunctionsEdit

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 transformationsEdit

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 S.A. Huq[3] 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  .

ReferencesEdit

  1. ^ a b c Adámek, Jiří; Herrlich, Horst; Strecker, George E. (1990). Abstract and Concrete Categories (PDF). John Wiley & Sons. ISBN 0-471-60922-6.
  2. ^ Rydheard, David E.; Burstall, Rod M. (1988). Computational category theory (PDF). Prentice Hall.
  3. ^ Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics 5 (2nd ed.), Springer-Verlag, p. 48, ISBN 0-387-98403-8

External linksEdit