nLab
category of elements

Idea

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

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 “Set-bundles;” see generalized universal bundle. The category of elements of $F$ is, in this sense, the Set-bundle classified by $F$. It comes equipped with a projection to $C$ which is a discrete fibration, and provides an equivalence between presheaves and discrete fibrations.

Forming a category of elements can be thought of as “unpacking” a concrete category. For example, consider a concrete category $C$ consisting of two objects $X,Y$ and two non-trivial morphisms $f,g$

The individual elements of $X,Y$ are “unpacked” and become objects of the new category. The “unpacked” morphisms are inherited in the obvious way from morphisms of $C$.

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 $\mathrm{Set}$, is called the Grothendieck construction.

Definition

Given a functor $P:C\to \mathrm{Set}$, the category of elements $\mathrm{el}(P)$ or ${\mathrm{El}}_P(C)$ (or obvious variations) may be understood in any of these equivalent ways:

• It is the category whose objects are pairs $(c,x)$ where $c$ is an object in $C$ and $x$ is an element in $P(c)$ and morphisms $(c,x)\to (c\prime ,x\prime )$ are morphisms $u:c\to c\prime$ such that $P(u)(x)=x\prime$.

• It is the pullback along $P$ of the universal Set-bundle $U:{\mathrm{Set}}_{*}\to \mathrm{Set}$

  $$\begin{array}{ccc}{\mathrm{El}}_P(C)& \to & {\mathrm{Set}}_{*}\\ {\downarrow }^{{\pi }_P}& & {\downarrow }^U\\ C& \to & \mathrm{Set}\end{array},$$\array{ El_P(C) &\to& Set_* \\ \downarrow^{\mathrlap{\pi_P}} && \downarrow^\mathrlap{U} \\ C &\to& Set }\,,

  where $U$ is the forgetful functor from pointed sets to sets.

• It is the comma category $(*/P)$, where $*$ is the inclusion of the one-point set $*:*\to \mathrm{Set}$ and $P:C\to \mathrm{Set}$ is itself:

  $$\begin{array}{ccc}{\mathrm{El}}_P(C)& \to & *\\ {\downarrow }^{{\pi }_P}& \Downarrow & {\downarrow }^{\mathrm{pt}}\\ C& \to & \mathrm{Set}\end{array}$$\array{ El_P(C) &\to& * \\ \downarrow^{\mathrlap{\pi_P}} &\Downarrow& \downarrow^{\mathrlap{pt}} \\ C &\to& Set }

• Its opposite is the comma category $(Y/P)$, where $Y$ is the Yoneda embedding $C^{\mathrm{op}}\to [C,\mathrm{Set}]$ and $P$ is the functor $*\to [C,\mathrm{Set}]$ which picks out $P$ itself:

  $$\begin{array}{ccc}{\mathrm{El}}_P(C{)}^{\mathrm{op}}& \stackrel{{\pi }_P^{\mathrm{op}}}{\to }& C^{\mathrm{op}}\\ \downarrow & \Downarrow & {\downarrow }^Y\\ *& \underset{P}{\to }& [C,\mathrm{Set}]\end{array}$$\array{ El_P(C)^{op} &\overset{\pi_P^{op}}{\to}& C^{op} \\ \downarrow &\Downarrow& \downarrow^{\mathrlap{Y}} \\ * & \underset{P}{\to}& [C,Set] }

  ${\mathrm{El}}_P(C)$ is also often written with coend notation as ${\int }^{C}P$, ${\int }^{c:C}P(c)$, or ${\int }^{c}P(c)$. This suggests the fact the set of objects of the category of elements is the disjoint union (sum) of all of the sets $P(c)$.

• It is the (strict) oplax colimit of the composite $C\stackrel{P}{\to }\mathrm{Set}\stackrel{\mathrm{disc}}{\to }\mathrm{Cat}$; see Grothendieck construction.

When $C$ is a concrete category and the functor $F:C\to \mathrm{Set}$ is simply the forgetful functor, we can define a functor

$$\mathrm{Explode}(-):={\mathrm{El}}_F(-).$$Explode(-) := 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 $\mathrm{el}:{\mathrm{Set}}^C\to \mathrm{Cat}$. This is perhaps most obvious when viewing it as an oplax colimit. Furthermore we have:

Theorem

The functor $\mathrm{el}:{\mathrm{Set}}^C\to \mathrm{Cat}$ is cocontinuous.

Proof

As remarked above, $\mathrm{el}$ is a strict weighted colimit 2-colimit, hence we have an isomorphism

$$\mathrm{el}(P)\cong {\int }^{c\in C}J(c)\times \mathrm{disc}(P(c))$$el(P) \cong \int^{c\in C} J(c) \times disc(P(c))

where the weight $J:C^{\mathrm{op}}\to \mathrm{Cat}$ is the functor $c\mapsto c/C$, and $\mathrm{disc}:\mathrm{Set}\hookrightarrow \mathrm{Cat}$ is the inclusion of the discrete categories. But since $\mathrm{disc}$ (regarded purely as a 1-functor) has a right adjoint (the functor which sends a -small- category $C$ into its set of elements $C_0$), it preserves (1-categorical) colimits. Since colimits also commute with colimits, the composite operation $el$ also preserves colimits.

Theorem

The functor $\mathrm{el}:{\mathrm{Set}}^C\to \mathrm{Cat}$ has a right adjoint (which is maybe a more direct way to see that it is cocontinuous).

Proof

By a simple coend computation

$$\begin{array}{rl}\mathrm{Cat}(\mathrm{el}(F),D)& \cong \mathrm{Cat}({\int }^{c}Jc\times \delta (Pc),D)\\ & \cong {\int }_c\mathrm{Cat}(Jc\times \delta (Fc),D)\\ & \cong {\int }_c\mathrm{Sets}(Fc,[Jc,D{]}_0)\\ & \cong {\mathrm{Set}}^C(F,K(D))\end{array}$$\begin{array}{rl} \mathbf{Cat}(el(F),D)&\cong \mathbf{Cat}\Big( \int^c J c\times\delta(P c), D\Big)\\ &\cong \int_c\mathbf{Cat}\big(J c\times \delta(F c),D\big)\\ &\cong \int_c \mathbf Sets\big(F c,[J c,D]_0\big)\\ &\cong \mathbf{Set}^{C}(F, K(D)) \end{array}

where $K(D):c\mapsto [Jc,D{]}_0$.

Now for any $C$, the terminal object of ${\mathrm{Set}}^C$ is the functor $\Delta 1$ constant at the point. The category of elements of $\Delta 1$ is easily seen to be just $C$ itself, so the unique transformation $P\to \Delta 1$ induces a projection functor ${\pi }_P:el(P)\to C$ defined by $(c,x)\mapsto c$ and $u\mapsto u$. The projection functor is a discrete opfibration, and can be viewed also as a $C$-indexed family of sets. When we regard $el(P)$ as equipped with ${\pi }_P$, we have an embedding of ${\mathrm{Set}}^C$ into $\mathrm{Cat}/C$.

Examples

Action Groupoid

In the case that $C=BG$ is the delooping groupoid of a group $G$, a functor $\varrho :BG\to \mathrm{Set}$ is a permutation representation $X$ of $G$ and its category of elements is the corresponding action groupoid $X//G$.

Proof

This is easily seen in terms of the characterization $\mathrm{el}(\varrho )\cong (*/\varrho )$, the category having as objects triples $(*,*;*\to \varrho (*)=X)$, namely elements of the set $X=\varrho (*)$, and as arrows $x\to y$ those $g\in BG$ such that

$$\begin{array}{ccc}*& \stackrel{x}{\to }& X\\ {}^1\downarrow & & {\downarrow }^g\\ *& \underset{y}{\to }& X\end{array}$$\array{ {*} & \overset{x}{\to} & X \\ {}^1\downarrow && \downarrow^g\\ {*} & \underset{y}{\to} & X }

commutes, namely $g.x=\varrho (g)(x)=y$. We can also present the right adjoint to $\mathrm{el}(-)$: one must consider the functor $J:B{G}^{\mathrm{op}}\to \mathrm{Cat}$, which represents $G$ in $\mathrm{Cat}$, and sends the unique object $*\in BG$ to $*/BG\cong G//G$, the left action groupoid of $G$. The functor $J$ sends $h\in G$ to an automorphism of $G//G$, obtained multiplying on the right $x\to gx$ to $xh\to xgh$.

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

Category of simplices

For a simplicial set regarded as a presheaf on the simplex category, the corresponding category of elements its called its category of simplices. See there for more.

Related concepts

• Grothendieck construction

• (∞,1)-Grothendieck construction