nLab
category of elements

Idea

For a functor $F:C\to \mathrm{Set}$, we may think of $\mathrm{Set}$ 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 analogue of the category of elements for functors landing in $\mathrm{Cat}$, rather than $\mathrm{Set}$, 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^{\pi_P} && \downarrow^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^{\pi_P} &\Downarrow& \downarrow^{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^{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.

Example: Action Groupoid

Given a vector space $V$, a group $G$, and a representation

$$\rho :\mathrm{BG}\to \mathrm{Vect},$$\rho:\mathbf{BG}\to Vect,

denote the image of $\rho$ by

$$V\nearrow G.$$V\nearrow G.

The action groupoid $V//G$ is then given by

$$V//G=\mathrm{Explode}(V\nearrow G).$$V//G=Explode(V\nearrow G).

References

• Wikipedia: Category of elements

• An Exercise in Groupoidification: The Path Integral

  • Exploding a Category
  • Library of Loops

Discussion

Eric: This seems like the gadget I’m trying to describe at Exploding a Category (I may remove that material once I understand what’s going on). I understand the concept (I think!), but why the notation ${\int }_{C}P$?

Toby: That notation is a reference to the standard notation for an end.

Urs: I’d say the integral sign here originates quite concretely in the idea that the category of elements of $F$ is the union of all the elements of $P(c)$ for all $c$.

Eric: I think this concept is important (for me anyway) and the intergal notation is cumbersome. What would be an acceptable alternative? For now, I think I will borrow from category of generalized elements and Wikipedia and denote it $\mathrm{El}(P)$. I can change it back if there is an uproar. I would actually prefer something like $\mathrm{Unpack}(C)$ to emphasize the relation to $C$.

PS: Don’t worry. I will make the edits once a nice notation is decided.

Toby: I don't really like $\mathrm{Unpack}$, although $\mathrm{El}$ seems fine. I do think that we should show the integral notation too, however, and give Urs's justification for it. (I'll do that now.)

Eric: Excellent. Instead of $\mathrm{El}$ (and forget about $\mathrm{Unpack}$), could we call it $\mathrm{Element}(P,C)$ or even ${\mathrm{El}}_P(C)$ and let $P:C\to \mathrm{Set}$? From what I can tell about Grothendieck construction, this would be more consistent. Lurie uses the notation $\mathrm{Groth}(P,C)$ for $P:C\to \mathrm{Cat}$ so ${\mathrm{El}}_P(C)$ with $P:C\to \mathrm{Set}$ makes sense to me.

Toby: Strictly speaking, $C$ is redundant, which is why people leave it out. I'm fine with having it in there, but I moved the very common $\mathrm{el}(P)$ higher up. I also dislike long words in mathematical notation, but ’$\mathrm{Elem}$’ is OK by me.

Mike: $\mathrm{el}(P)$ is a fairly common notation.

Eric: Should the above statement be changed from disjoint union to direct product? - Eric

Mike: No. Each object of $\mathrm{el}(P)$ is an element of exactly one $P(c)$, so the set of objects is the disjoint union. An element of the direct product would consist of an element of $P(c)$ for every $c$.

Toby: Even as a coend, it's still like an integral: a sum.

Eric: Thanks. I hope my changes are acceptable. I’m pretty happy with the article now. I also added a note about $\mathrm{el}(P)$.