nLab category of elements




The category of elements of a functor F:π’žβ†’F : \mathcal{C} \to Set is a category el(F)β†’π’žel(F) \to \mathcal{C} sitting over the domain category π’ž\mathcal{C}, such that the fiber over an object cβˆˆπ’žc \in \mathcal{C} is the set F(c)F(c).

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 β€œSet-bundles;” see generalized universal bundle. The category of elements of FF is, in this sense, the Set-bundle classified by FF. It comes equipped with a projection to π’ž\mathcal{C} which is a discrete opfibration, and provides an equivalence between Set\mathbf{Set}-valued functors and discrete opfibrations. (There is a dual category of elements that applies to contravariant Set\mathbf{Set}-valued functors, i.e. presheaves, and produces discrete fibrations.)

Forming a category of elements can be thought of as β€œunpacking” a concrete category. For example, consider a concrete category π’ž\mathcal{C} consisting of two objects X,YX,Y and two non-trivial morphisms f,gf,g

The individual elements of X,YX,Y are β€œunpacked” and become objects of the new category. The β€œunpacked” morphisms are inherited in the obvious way from morphisms of π’ž\mathcal{C}.

Note that an β€œunpacked” category of elements can be β€œrepackaged”.

The generalization of the category of elements for functors landing in Cat, rather than just Set\mathbf{Set}, is called the Grothendieck construction.


Given a functor F:π’žβ†’SetF:\mathcal{C}\to\mathbf{Set}, the category of elements el(F)el(F) or El F(π’ž)El_F(\mathcal{C}) (or obvious variations) may be understood in any of these equivalent ways:

  • It is the category whose objects are pairs (c,x)(c,x) where cc is an object in π’ž\mathcal{C} and xx is an element in F(c)F(c) and morphisms (c,x)β†’(cβ€²,xβ€²)(c,x)\to(c',x') are morphisms u:cβ†’cβ€²u:c\to c' such that F(u)(x)=xβ€²F(u)(x) = x'.

  • It is the pullback along FF of the universal Set-bundle U:Set *β†’SetU : \mathbf{Set}_* \to \mathbf{Set}

    where UU is the forgetful functor from pointed sets to sets.

  • It is the comma category (*/F)(*/F), where ** is the inclusion of the one-point set *:*β†’Set*:*\to \mathbf{Set} and F:π’žβ†’SetF:\mathcal{C}\to \mathbf{Set} is itself:

  • Its opposite is the comma category (Y/F)(Y/F), where YY is the Yoneda embedding π’ž opβ†’[π’ž,Set]\mathcal{C}^{op}\to [\mathcal{C},\mathbf{Set}] and FF is the functor *β†’[π’ž,Set]*\to [\mathcal{C},\mathbf{Set}] which picks out FF itself:

El F(π’ž)El_F(\mathcal{C}) is also often written with coend notation as ∫ π’žF\int^\mathcal{C} F, ∫ c:π’žF(c)\int^{c: \mathcal{C}} F(c), or ∫ cF(c)\int^c F(c). This suggests the fact the set of objects of the category of elements is the disjoint union (sum) of all of the sets F(c)F(c).

When π’ž\mathcal{C} is a concrete category and the functor F:π’žβ†’SetF:\mathcal{C}\to \mathbf{Set} is simply the forgetful functor, we can define a functor

Explode(βˆ’):=El F(βˆ’).Explode(-) := El_F(-).

This is intended to illustrate the concept that constructing a category of elements is like β€œunpacking” or β€œexploding” a category into its elements.


The category of elements defines a functor el:Set π’žβ†’Catel : \mathbf{Set}^{\mathcal{C}} \to \mathbf{Cat}. This is perhaps most obvious when viewing it as an oplax colimit. Furthermore we have:


The functor el:Set π’žβ†’Catel : \mathbf{Set}^{\mathcal{C}} \to \mathbf{Cat} is cocontinuous.


As remarked above, elel is a strict weighted 2-colimit, hence we have an isomorphism

el(F)β‰…βˆ« cβˆˆπ’žJ(c)Γ—disc(F(c)) el(F) \cong \int^{c\in \mathcal{C}} J(c) \times disc(F(c))

where the weight J:π’ž opβ†’CatJ:\mathcal{C}^{op} \to \mathbf{Cat} is the functor c↦c/π’žc\mapsto c/\mathcal{C}, and disc:Setβ†ͺCatdisc:\mathbf{Set}\hookrightarrow \mathbf{Cat} is the inclusion of the discrete categories. But since discdisc (regarded purely as a 1-functor) has a right adjoint (the functor which sends a -small- category π’ž\mathcal{C} into its set of elements π’ž 0\mathcal{C}_0), it preserves (1-categorical) colimits. Since colimits also commute with colimits, the composite operation el\el also preserves colimits.


The functor el:Set π’žβ†’Catel\colon \mathbf{Set}^{\mathcal{C}} \to \mathbf{Cat} has a right adjoint (which is maybe a more direct way to see that it is cocontinuous).


By a simple coend computation

Cat(el(F),D) β‰…Cat(∫ cJcΓ—Ξ΄(Fc),D) β‰…βˆ« cCat(JcΓ—Ξ΄(Fc),D) β‰…βˆ« cSet(Fc,[Jc,D] 0) β‰…Set π’ž(F,K(D)) \begin{array}{rl} \mathbf{Cat}(el(F),D)&\cong \mathbf{Cat}\Big( \int^c J c\times\delta(F c), D\Big)\\ &\cong \int_c\mathbf{Cat}\big(J c\times \delta(F c),D\big)\\ &\cong \int_c \mathbf{Set}\big(F c,[J c,D]_0\big)\\ &\cong \mathbf{Set}^{\mathcal{C}}(F, K(D)) \end{array}

where K(D):c↦[Jc,D] 0K(D)\colon c\mapsto [J c,D]_0.

Now for any π’ž\mathcal{C}, the terminal object of Set π’ž\mathbf{Set}^\mathcal{C} is the functor Ξ”1\Delta 1 constant at the point. The category of elements of Ξ”1\Delta 1 is easily seen to be just π’ž\mathcal{C} itself, so the unique transformation Fβ†’Ξ”1F\to \Delta 1 induces a projection functor Ο€ F:el(F)β†’π’ž\pi_F: \el(F) \to \mathcal{C} defined by (c,x)↦c(c,x)\mapsto c and u↦uu\mapsto u. The projection functor is a discrete opfibration, and can be viewed also as a π’ž\mathcal{C}-indexed family of sets. When we regard el(F)\el(F) as equipped with Ο€ F\pi_F, we have an embedding of Set π’ž\mathbf{Set}^\mathcal{C} into Cat/π’ž\mathbf{Cat}/\mathcal{C}.

Note that the canonical projection El(F)β†’C\operatorname{El}(F) \to \mathbf{C} is not usually full. For example, let Bβ„•\mathbf{B}\mathbb{N} be the one-object category which carries the monoid (β„•,+)(\mathbb{N}, +) as its endomorphism monoid, and let FF be the action of (β„•,+)(\mathbb{N}, +) on the set β„•\mathbb{N} by n.m=m+nn.m = m + n. Then the image of any hom-set between k,kβ€²k, k' is a subsingleton subset of β„•\mathbb{N}.

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 00 or 11, while objects in the groupoid might have nontrivial automorphism groups.


Representable Presheaves

Let Y(c):π’ž opβ†’SetY(c):\mathcal{C}^{op}\to \mathbf{Set} be a representable presheaf with Y(c)(d)=Hom π’ž(d,c)Y(c)(d)=Hom_{\mathcal{C}}(d,c). Consider the contravariant category of elements ∫ π’žY(c)\int_\mathcal{C} Y(c) . This has objects (d 1,p 1)(d_1,p_1) with p 1∈Y(c)(d 1)p_1\in Y(c)(d_1), hence p 1p_1 is just an arrow d 1β†’cd_1\to c in π’ž\mathcal{C}. A map from (d 1,p 1)(d_1, p_1) to (d 2,p 2)(d_2, p_2) is just a map u:d 1β†’d 2u:d_1\to d_2 such that p 2∘u=p 1p_2\circ u =p_1 but this is just a morphism from p 1p_1 to p 2p_2 in the slice category π’ž/c\mathcal{C}/c. Accordingly we see that ∫ π’žY(c)β‰ƒπ’ž/c\int_\mathcal{C} Y(c)\simeq \mathcal{C}/c .

This equivalence comes in handy when one wants to compute slices of presheaf toposes over representable presheaves Y(c)Y(c) since PSh(∫ π’žF)≃PSh(π’ž)/FPSh(\int_\mathcal{C} F) \simeq PSh(\mathcal{C})/F in general for presheaves F:π’ž opβ†’SetF:\mathcal{C}^{op}\to \mathbf{Set} , whence PSh(π’ž)/Y(c)≃PSh(π’ž/c)PSh(\mathcal{C})/Y(c) \simeq PSh(\mathcal{C}/c) . An instructive example of this construction is spelled out in detail at hypergraph.

Action Groupoid

In the case that π’ž=BG\mathcal{C} = \mathbf{B}G is the delooping groupoid of a group GG, a functor Ο±:BGβ†’Set\varrho : \mathbf{B}G \to \mathbf{Set} is a permutation representation XX of GG and its category of elements is the corresponding action groupoid X//GX/\!/G.


This is easily seen in terms of the characterization el(Ο±)β‰…(*/Ο±)el(\varrho)\cong (*/\varrho), the category having as objects triples (*,*;*β†’Ο±(*)=X)(*,*; *\to \varrho(*)=X), namely elements of the set X=Ο±(*)X=\varrho(*), and as arrows xβ†’yx\to y those g∈BGg\in \mathbf{B}G such that

commutes, namely g.x=Ο±(g)(x)=yg . x=\varrho(g)(x)=y. We can also present the right adjoint to el(βˆ’)el(-): one must consider the functor J:BG opβ†’CatJ\colon \mathbf{B}G^{op}\to \mathbf{Cat}, which represents GG in Cat\mathbf{Cat}, and sends the unique object *∈BG*\in \mathbf{B}G to */BGβ‰…G//G*/\mathbf{B}G\cong G/\!/G, the left action groupoid of GG. The functor JJ sends h∈Gh\in G to an automorphism of G//GG/\!/G, obtained multiplying on the right xβ†’gxx\to g x to xhβ†’xghx h\to x g h.

Now for any category DD, K(D)(*)K( D)(*) is exactly the set of functors [G//G,D][G/\!/G, D], which inherits from G//GG/\!/G an obvious action: given F∈[G//G,D]F\in [G/\!/G, D] we define F h=J(h) *F=F∘J(h):g↦F(gh)F^h=J(h)^*F=F \circ J(h) \colon g \mapsto F(g h).

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. 12 of

Last revised on January 30, 2024 at 05:38:50. See the history of this page for a list of all contributions to it.