Given a collection of “parameterized objects”, i.e. a functor , it is often of interest to consider the category whose objects are generalized elements of the objects of in the image of , and whose morphisms are the maps between these generalized elements induced by the value of on morphisms in .
For Set and and with generalized element read as “ordinary element of a set” is yields the category of elements of the (co)presheaf .
Moreover, the description of of the category of elements of a presheaf in terms of a pullback of a generalized universal bundle generalizes directly to categories of generalized elements.
Let be a pointed object in Cat, i.e. a category equipped with a choice of one of its objects.
Recall that a morphism in may be called a generalized element in “with domain of definition” being the object .
For instance if Set the canonical choice is the set with a single element. Generalized elements of a set “with domain of definition” are just the ordinary elements of a set.
Notice that the over category is the category of generalized elements of with domain of definition :
objects are such generalized elements of objects ;
morphisms are given whenever a morphism in takes the element to , i.e. whenever there is a commuting triangle
Notice that the canonical projection from the over category that forgets the tip of these trangles may be regarded as the generalized universal bundle for the given pointed category : it is the left composite vertical morphism in the pullback
(see also comma category for more on this perspective). So in fact such “categories of generalized elements” are precisely the generalized universal bundles in the 1-categorical context. And both are really fundamentally to be thought of as intermediate steps in the computation of weak pullbacks, as described now.
The above allows to generalize the notion of category of generalized elements a bit further to that of generalized elements of functors with values in : let be a functor with codomain our category with point .
The category of generalized elements of is the pullback
the objects of are all the generalized elements for all ;
a morphism between two such generalized elements is a commuting triangle
for all morphisms in .
ordinary category of elements
For Set and the above reproduces the notion of category of elements# of a presheaf.
Given a vector space , a group recall that a representation of on
is canonically identified with a functor
The category Vect of -vector spaces for some field has a standard point , namely the field itself, regarded as the canonical 1-dimensional -vector space over itself.
The corresponding over category of generalized elements of Vect has as objects pointed vector spaces and as morphisms linear maps of pointed vector spaces that map the chosen vectors to each other.
Now, as described in detail at action groupoid the category of generalized elements of the representation is the action groupoid of acting on
As described there, is the groupoid incarnation of the vector bundle that is associated via to the universal -bundle on the classifying space .