# Idea

Given a collection of “parameterized objects”, i.e. a functor $F : C \to D$, it is often of interest to consider the category whose objects are generalized elements of the objects of $D$ in the image of $F$, and whose morphisms are the maps between these generalized elements induced by the value of $F$ on morphisms in $C$.

For $D =$ Set and and with generalized element read as “ordinary element of a set” is yields the category of elements of the (co)presheaf $F : C \to D$.

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.

# Definition

Let $D$ be a pointed object in Cat, i.e. a category equipped with a choice $pt_D : {*} \to D$ of one of its objects.

Recall that a morphism $pt_D \to d$ in $D$ may be called a generalized element in $D$ “with domain of definition” being the object $pt_D$.

For instance if $D =$ Set the canonical choice is $pt_{Set} = {*}$ 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 $({pt_D/D})$ is the category of generalized elements of $D$ with domain of definition $pt_D$:

• objects are such generalized elements $\delta : pt_D \to d$ of objects $d \in D$;

• morphisms $\delta \to \gamma$ are given whenever a morphism $f : d \to d'$ in $D$ takes the element $\delta$ to $\delta'$, i.e. whenever there is a commuting triangle

$\array{ && pt_D \\ & {}^\delta\swarrow && \searrow^{\delta'} \\ d &&\stackrel{f}{\to}&& d' } \,.$

Notice that the canonical projection $(pt_D/D) \to D$ from the over category that forgets the tip of these trangles may be regarded as the generalized universal bundle for the given pointed category $D$: it is the left composite vertical morphism in the pullback

$\array{ (pt_D/D) &\to& {*} \\ \downarrow && \downarrow \\ D^I &\stackrel{d_0}{\to}& D \\ \downarrow^{d_1} \\ D }$

(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 $D$: let $F : C \to D$ be a functor with codomain our category $D$ with point $pt_D$.

The category of generalized elements of $F$ is the pullback $El_{pt_D}(F) := C \times_D (pt_D/D)$

$\array{ El_{pt_D}(F) &\to& (pt_D/D) \\ \downarrow && \downarrow \\ C &\stackrel{F}{\to}& D } \,.$

This means:

• the objects of $El_{pt_D}(F)$ are all the generalized elements $\delta_c : pt_D \to F(c)$ for all $c \in C$;

• a morphism $\delta_c \to \delta_{c'}$ between two such generalized elements is a commuting triangle

$\array{ && pt_D \\ & {}^{\delta_c}\swarrow && \searrow^{\delta_{c'}} \\ F(c) && \stackrel{F(f)}{\to} && F(c') } \,.$

for all morphisms $f : c \to c'$ in $C$.

# Examples

## ordinary category of elements

For $D =$ Set and $pt_{Set} = {*}$ the above reproduces the notion of category of elements of a presheaf.

## Action Groupoid

Given a vector space $V$, a group $G$ recall that a representation of $G$ on $V$

$V\bullet\righttoleftarrow G$

is canonically identified with a functor

$\rho : \mathbf{B} G \to Vect \,.$
$\rho : ({*} \stackrel{g}{\to} {*}) \mapsto (V \stackrel{\rho(g)}{\to} V) \,.$

The category Vect of $k$-vector spaces for some field $k$ has a standard point $pt_{Vect} \to Vect$, namely the field $k$ itself, regarded as the canonical 1-dimensional $k$-vector space over itself.

The corresponding over category of generalized elements of Vect $(pt_{Vect}/ Vect)$ 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 $\rho$ is the action groupoid $V//G$ of $G$ acting on $V$

$\array{ V//G &\to& Vect_* \\ \downarrow && \downarrow \\ \mathbf{B} G &\to& Vect } \,.$

As described there, $V//G \to \mathbf{B}G$ is the groupoid incarnation of the vector bundle that is associated via $\rho$ to the universal $G$-bundle on the classifying space $B G$.