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category of elements (changes) in nLab

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Idea

The category of elements of a functor F:𝒞→F : \mathcal{C} \to Set is a category el(F)→𝒞el(F) \to \mathcal{C} sitting over the domain category 𝒞\mathcal{C}, such that the fiber over an object c∈𝒞c \in \mathcal{C} is the set F(c)F(c).

This is a special case of the Grothendieck construction for covariant functors, by considering sets as discrete categories. There is a similar special case of the Grothendieck construction for contravariant functors which takes a functor F:𝒞 op→SetF:\mathcal{C}^{op} \to \mathrm{Set} to a category over 𝒞\mathcal{C}. However, this article is only about the covariant case.

We may think of Set as the classifying space of “Set-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 𝒞\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 “unpacking” a concrete category. For example, consider a concrete category 𝒞\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 “unpacked” and become objects of the new category. The “unpacked” morphisms are inherited in the obvious way from morphisms of 𝒞\mathcal{C}.

Note that an “unpacked” category of elements can be “repackaged”.

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

Definition

Given a functor F:𝒞→SetF:\mathcal{C}\to\mathbf{Set}, the category of elements el(F)el(F) or El F(𝒞)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 𝒞\mathcal{C} and xx is an element in F(c)F(c) and morphisms (c,x)→(c′,x′)(c,x)\to(c',x') are morphisms u:c→c′u:c\to c' such that F(u)(x)=x′F(u)(x) = x'.

  • It is the pullback along FF of the universal Set-bundle U:Set *→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 *:*→Set*:*\to \mathbf{Set} and F:𝒞→SetF:\mathcal{C}\to \mathbf{Set} is itself:

\begin{center} \begin{tikzcd} \text{El}_F(\mathcal{C}) \arrow[r, ] \arrow[d, \pi_F] & \mathbf{Set}_* \arrow[d, U] \ \mathcal{C} \arrow[r, ] & \mathbf{Set} \end{tikzcd} \end{center}

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

\begin{center} \begin{tikzcd} \text{El}_F(\mathcal{C})^{op} \arrow[r, \pi_F^{op}] \arrow[d, ] & \mathcal{C}^{op} \arrow[d, U] \ * \arrow[r, F] & {[\mathcal{C}, \mathbf{Set}]} \end{tikzcd} \end{center}

El F(𝒞)El_F(\mathcal{C}) is also often written with coend notation as ∫ 𝒞F\int^\mathcal{C} F, ∫ c:𝒞F(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 𝒞\mathcal{C} is a concrete category and the functor F:𝒞→SetF:\mathcal{C}\to \mathbf{Set} is simply the forgetful functor, we can define a functor

Explode(−) : ≔=El F(−). Explode(-) := \coloneqq El_F(-). El_F(-) \,.

This is intended to illustrate the concept that constructing a category of elements is like “unpacking” or “exploding” a category into its elements.

Properties

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

Theorem

The functor el:Set 𝒞→Catel : \mathbf{Set}^{\mathcal{C}} \to \mathbf{Cat} is cocontinuous.

Proof

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

el(F)≅∫ c∈𝒞J(c)×disc(F(c)), el(F) \cong \;\cong\; \int^{c\in \mathcal{C}} J(c) \times disc(F(c)) disc\big(F(c)\big) \,,

where the weight J:𝒞 op→CatJ:\mathcal{C}^{op} \to \mathbf{Cat} is the functor c↦c/𝒞c\mapsto c/\mathcal{C}, and disc:Set↪Catdisc:\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 𝒞\mathcal{C} into its set of elements 𝒞 0\mathcal{C}_0), it preserves (1-categorical) colimits. Since colimits also commute with colimits, the composite operation el\el also preserves colimits.

Theorem

The functor el:Set 𝒞→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).

Proof

By a simple coend computation:

Cat(el(F),D) ≅Cat(∫ cJc×δ(Fc),D) ≅∫ cCat(Jc×δ(Fc),D) ≅∫ cSet(Fc,[Jc,D] 0) ≅Set 𝒞(F,K(D)) \begin{aligned} \mathbf{Cat}(el(F),D)&\cong \mathbf{Cat}\left( \int^c J c\times\delta(F c), D\right)\\ &\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{aligned}

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

Now for any 𝒞\mathcal{C}, the terminal object of Set 𝒞\mathbf{Set}^\mathcal{C} is the functor Δ1\Delta 1 constant at the point. The category of elements of Δ1\Delta 1 is easily seen to be just 𝒞\mathcal{C} itself, so the unique transformation F→Δ1F\to \Delta 1 induces a projection functor π F:el(F)→𝒞\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 𝒞\mathcal{C}-indexed family of sets. When we regard el(F)\el(F) as equipped with π F\pi_F, we have an embedding of Set 𝒞\mathbf{Set}^\mathcal{C} into Cat/𝒞\mathbf{Cat}/\mathcal{C}.

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

Examples

Representable Presheaves

Let Y(c):𝒞 op→SetY(c):\mathcal{C}^{op}\to \mathbf{Set} be a representable presheaf with Y(c)(d)=Hom 𝒞(d,c)Y(c)(d)=Hom_{\mathcal{C}}(d,c). Consider the contravariant category of elements ∫ 𝒞Y(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 1→cd_1\to c in 𝒞\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 1→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 𝒞/c\mathcal{C}/c. Accordingly we see that ∫ 𝒞Y(c)≃𝒞/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(∫ 𝒞F)≃PSh(𝒞)/FPSh(\int_\mathcal{C} F) \simeq PSh(\mathcal{C})/F in general for presheaves F:𝒞 op→SetF:\mathcal{C}^{op}\to \mathbf{Set} , whence PSh(𝒞)/Y(c)≃PSh(𝒞/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 𝒞=BG\mathcal{C} = \mathbf{B}G is the delooping groupoid of a group GG, a functor ϱ:BG→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.

Proof

This is easily seen in terms of the characterization el(ϱ)≅(*/ϱ)el(\varrho)\cong (*/\varrho), the category having as objects triples (*,*;*→ϱ(*)=X)(*,*; *\to \varrho(*)=X), namely elements of the set X=ϱ(*)X=\varrho(*), and as arrows x→yx\to y those g∈BGg\in \mathbf{B}G such that \begin{center} \begin{tikzcd} * \arrow[r, x] \arrow[d, 1] & X \arrow[d, g] \ * \arrow[r, y] & X \end{tikzcd} \end{center} commutes, namely g.x=ϱ(g)(x)=yg . x=\varrho(g)(x)=y. We can also present the right adjoint to el(−)el(-): one must consider the functor J:BG op→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 */BG≅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 x→gxx\to g x to xh→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.

Reference

A very nice introduction emphasizing the connections to monoid theory is ch. 12 of

Last revised on February 18, 2025 at 13:39:37. See the history of this page for a list of all contributions to it.