Comma 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 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).
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.
Suppose that formula_1, formula_2, and formula_3 are categories, and formula_4 and formula_5 (for source and target) are functors:
formula_6
We can form the comma category formula_7 as follows:
Morphisms are composed by taking formula_22 to be formula_23, whenever the latter expression is defined. The identity morphism on an object formula_8 is formula_25.
Slice category.
The first special case occurs when formula_26, the functor formula_4 is the identity functor, and formula_28 (the category with one object formula_29 and one morphism). Then formula_30 for some object formula_31 in formula_1.
formula_33
In this case, the comma category is written formula_34, and is often called the "slice category" over formula_31 or the category of "objects over formula_31". The objects formula_37 can be simplified to pairs formula_38, where formula_39. Sometimes, formula_40 is denoted by formula_41. A morphism formula_42 from formula_43 to formula_44 in the slice category can then be simplified to an arrow formula_18 making the following diagram commute:
Coslice category.
The dual concept to a slice category is a coslice category. Here, formula_46, formula_4 has domain formula_48 and formula_5 is an identity functor.
formula_50
In this case, the comma category is often written formula_51, where formula_52 is the object of formula_2 selected by formula_4. It is called the "coslice category" with respect to formula_55, or the category of "objects under formula_55". The objects are pairs formula_57 with formula_58. Given formula_57 and formula_60, a morphism in the coslice category is a map formula_19 making the following diagram commute:
Arrow category.
formula_4 and formula_5 are identity functors on formula_3 (so formula_65).
formula_66
In this case, the comma category is the arrow category formula_67. Its objects are the morphisms of formula_3, and its morphisms are commuting squares in formula_3.
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 formula_5 is the forgetful functor mapping an abelian group to its underlying set, and formula_71 is some fixed set (regarded as a functor from 1), then the comma category formula_72 has objects that are maps from formula_71 to a set underlying a group. This relates to the left adjoint of formula_5, 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 formula_72 is the canonical injection formula_76, where formula_77 is the free group generated by formula_71.
An object of formula_72 is called a "morphism from formula_71 to formula_5" or a "formula_5-structured arrow with domain formula_71". An object of formula_84 is called a "morphism from formula_4 to formula_86" or a "formula_4-costructured arrow with codomain formula_86".
Another special case occurs when both formula_4 and formula_5 are functors with domain formula_48. If formula_92 and formula_93, then the comma category formula_7, written formula_95, is the discrete category whose objects are morphisms from formula_9 to formula_11.
An inserter category is a (non-full) subcategory of the comma category where formula_98 and formula_99 are required. The comma category can also be seen as the inserter of formula_100 and formula_101, where formula_102 and formula_103 are the two projection functors out of the product category formula_104.
Properties.
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.
Limits and colimits in comma categories may be "inherited". If formula_1 and formula_2 are complete, formula_152 is a continuous functor, and formula_153 is another functor (not necessarily continuous), then the comma category formula_7 produced is complete, and the projection functors formula_155 and formula_156 are continuous. Similarly, if formula_1 and formula_2 are cocomplete, and formula_159 is cocontinuous, then formula_7 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 formula_3 be a category with formula_162 the functor taking each object formula_163 to formula_164 and each arrow formula_134 to formula_166. A universal morphism from formula_167 to formula_168 consists, by definition, of an object formula_164 and morphism formula_170 with the universal property that for any morphism formula_171 there is a unique morphism formula_172 with formula_173. In other words, it is an object in the comma category formula_174 having a morphism to any other object in that category; it is initial. This serves to define the coproduct in formula_3, when it exists.
Adjunctions.
Lawvere showed that the functors formula_176 and formula_177 are adjoint if and only if the comma categories formula_178 and formula_179, with formula_180 and formula_181 the identity functors on formula_182 and formula_3 respectively, are isomorphic, and equivalent elements in the comma category can be projected onto the same element of formula_184. 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 formula_185 are equal, then the diagram which defines morphisms in formula_186 with formula_187 is identical to the diagram which defines a natural transformation formula_188. The difference between the two notions is that a natural transformation is a particular collection of morphisms of type of the form formula_189, 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
that a natural transformation formula_190, with formula_191, corresponds to a functor formula_192 which maps each object formula_9 to formula_194 and maps each morphism formula_195 to formula_17. This is a bijective correspondence between natural transformations formula_188 and functors formula_192 which are sections of both forgetful functors from formula_186.