category of generalized elements


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

For D=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:CDF : 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.


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

Recall that a morphism pt Ddpt_D \to d in DD may be called a generalized element in DD “with domain of definition” being the object pt Dpt_D.

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

  • objects are such generalized elements δ:pt Dd\delta : pt_D \to d of objects dDd \in D;

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

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

Notice that the canonical projection (pt D/D)D(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 DD: it is the left composite vertical morphism in the pullback

(pt D/D) * D I d 0 D d 1 D \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 DD: let F:CDF : C \to D be a functor with codomain our category DD with point pt Dpt_D.

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

El pt D(F) (pt D/D) C F 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)El_{pt_D}(F) are all the generalized elements δ c:pt DF(c)\delta_c : pt_D \to F(c) for all cCc \in C;

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

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

    for all morphisms f:ccf : c \to c' in CC.


ordinary category of elements

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

Action Groupoid

Given a vector space VV, a group GG recall that a representation of GG on VV

VGV\bullet\righttoleftarrow G

is canonically identified with a functor

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

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

The corresponding over category of generalized elements of Vect (pt Vect/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 ρ\rho is the action groupoid V//GV//G of GG acting on VV

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

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


Last revised on August 12, 2019 at 07:19:25. See the history of this page for a list of all contributions to it.