For a functor , we may think of 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 analogue of the category of elements for functors landing in , rather than , 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 .
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.
Given a vector space , a group , and a representation
denote the image of by
The action groupoid is then given by
Eric: This seems like the gadget I’m trying to describe at Exploding a Category (I may remove that material once I understand what’s going on). I understand the concept (I think!), but why the notation ?
Toby: That notation is a reference to the standard notation for an end.
Urs: I’d say the integral sign here originates quite concretely in the idea that the category of elements of is the union of all the elements of for all .
Eric: I think this concept is important (for me anyway) and the intergal notation is cumbersome. What would be an acceptable alternative? For now, I think I will borrow from category of generalized elements and Wikipedia and denote it . I can change it back if there is an uproar. I would actually prefer something like to emphasize the relation to .
PS: Don’t worry. I will make the edits once a nice notation is decided.
Toby: I don't really like , although seems fine. I do think that we should show the integral notation too, however, and give Urs's justification for it. (I'll do that now.)
Eric: Excellent. Instead of (and forget about ), could we call it or even and let ? From what I can tell about Grothendieck construction, this would be more consistent. Lurie uses the notation for so with makes sense to me.
Toby: Strictly speaking, is redundant, which is why people leave it out. I'm fine with having it in there, but I moved the very common higher up. I also dislike long words in mathematical notation, but ’’ is OK by me.
Mike: is a fairly common notation.
Eric: Should the above statement be changed from disjoint union to direct product? - Eric
Mike: No. Each object of is an element of exactly one , so the set of objects is the disjoint union. An element of the direct product would consist of an element of for every .
Toby: Even as a coend, it's still like an integral: a sum.
Eric: Thanks. I hope my changes are acceptable. I’m pretty happy with the article now. I also added a note about .