The category of elements of a functor Set is a category sitting over the domain category , such that the fiber over an object is the set .
This is a special case of the Grothendieck construction, by considering sets as discrete categories.
We may think of Set as the classifying space of “Set-bundles;” see generalized universal bundle. The category of elements of is, in this sense, the Set-bundle classified by . It comes equipped with a projection to which is a discrete fibration, and provides an equivalence between presheaves and discrete fibrations.
Forming a category of elements can be thought of as “unpacking” a concrete category. For example, consider a concrete category consisting of two objects and two non-trivial morphisms
The individual elements of are “unpacked” and become objects of the new category. The “unpacked” morphisms are inherited in the obvious way from morphisms of .
Note that an “unpacked” category of elements can be “repackaged”.
The generalization of the category of elements for functors landing in Cat, rather than just , is called the Grothendieck construction.
Given a functor , the category of elements or (or obvious variations) may be understood in any of these equivalent ways:
It is the category whose objects are pairs where is an object in and is an element in and morphisms are morphisms such that .
It is the pullback along of the universal Set-bundle
where is the forgetful functor from pointed sets to sets.
It is the comma category , where is the inclusion of the one-point set and is itself:
Its opposite is the comma category , where is the Yoneda embedding and is the functor which picks out itself:
is also often written with coend notation as , , or . This suggests the fact the set of objects of the category of elements is the disjoint union (sum) of all of the sets .
It is the (strict) oplax colimit of the composite ; see Grothendieck construction.
When is a concrete category and the functor is simply the forgetful functor, we can define a functor
This is intended to illustrate the concept that constructing a category of elements is like “unpacking” or “exploding” a category into its elements.
The category of elements defines a functor . This is perhaps most obvious when viewing it as an oplax colimit. Furthermore we have:
The functor is cocontinuous.
As remarked above, is a strict weighted colimit 2-colimit, hence we have an isomorphism
where the weight is the functor , and is the inclusion of the discrete categories. But since (regarded purely as a 1-functor) has a right adjoint (the functor which sends a -small- category into its set of elements ), it preserves (1-categorical) colimits. Since colimits also commute with colimits, the composite operation also preserves colimits.
The functor has a right adjoint (which is maybe a more direct way to see that it is cocontinuous).
By a simple coend computation
Now for any , the terminal object of is the functor constant at the point. The category of elements of is easily seen to be just itself, so the unique transformation induces a projection functor defined by and . The projection functor is a discrete opfibration, and can be viewed also as a -indexed family of sets. When we regard as equipped with , we have an embedding of into .
Note that while the canonical projection is surjective on objects, it is not usually full. For example, let be the one-object category which carries the monoid as its endomorphism monoid, and let be the action of on the set by . Then the image of any hom-set between is a singleton subset of .
More generally, the universal covering groupoid of a groupoid is just the category of elements of its action on itself by composition. Since this action is faithful and transitive, hom-sets in the category of elements are always or , while objects in the groupoid might have nontrivial automorphism groups.
Let be a representable presheaf with . Consider the contravariant category of elements . This has objects with , hence is just an arrow in . A map from to is just a map such that but this is just a morphism from to in the slice category . Accordingly we see that .
This equivalence comes in handy when one wants to compute slices of presheaf toposes over representable presheaves since in general for presheaves , whence . An instructive example of this construction is spelled out in detail at hypergraph.
In the case that is the delooping groupoid of a group , a functor is a permutation representation of and its category of elements is the corresponding action groupoid .
This is easily seen in terms of the characterization , the category having as objects triples , namely elements of the set , and as arrows those such that
commutes, namely . We can also present the right adjoint to : one must consider the functor , which represents in , and sends the unique object to , the left action groupoid of . The functor sends to an automorphism of , obtained multiplying on the right to .
Now for any category , is exactly the set of functors , which inherits from an obvious action: given we define .
Category of simplices
For a simplicial set regarded as a presheaf on the simplex category, the corresponding category of elements is called its category of simplices. See there for more.
A very nice introduction emphasizing the connections to monoid theory is ch. 11 of