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.

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 $P(c)$.

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 is naturally equipped with a projection functor ${\pi }_P:{\mathrm{El}}_P(C)\to C$ given by $(c,x)\mapsto c$ and $u\mapsto u$. This projection is a discrete opfibration and can be viewed also as a $C$-indexed family of sets.

Examples

Action Groupoid

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

Related concepts

• category of elements

• Grothendieck construction

• (∞,1)-Grothendieck construction