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\begin{document}

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\section*{category of elements}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{representable_presheaves}{Representable Presheaves}\dotfill \pageref*{representable_presheaves} \linebreak
\noindent\hyperlink{action_groupoid}{Action Groupoid}\dotfill \pageref*{action_groupoid} \linebreak
\noindent\hyperlink{category_of_simplices}{Category of simplices}\dotfill \pageref*{category_of_simplices} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{reference}{Reference}\dotfill \pageref*{reference} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{category of elements} of a [[functor]] $F : C \to$ [[Set]] is a [[category]] $el(F) \to C$ 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 opfibration]], and provides an equivalence between $Set$-valued functors and discrete opfibrations. (There is a dual category of elements that applies to contravariant $Set$-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 $C$ 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 $Set$, is called the [[Grothendieck construction]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

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

\begin{itemize}%
\item 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 $(c,x)\to(c',x')$ are morphisms $u:c\to c'$ such that $P(u)(x) = x'$.


\item It is the [[pullback]] along $P$ of the [[generalized universal bundle|universal Set-bundle]] $U : Set_* \to Set$

\begin{displaymath}
\itexarray{ 
     El_P(C) &\to& Set_* 
     \\ 
     \downarrow^{\mathrlap{\pi_P}} && 
     \downarrow^\mathrlap{U} 
     \\ 
     C &\to& Set
   }\,,
\end{displaymath}
where $U$ is the [[forgetful functor]] from [[pointed set|pointed sets]] to sets.


\item It is the [[comma category]] $(*/P)$, where $*$ is the inclusion of the one-point set $*:*\to Set$ and $P:C\to Set$ is itself:

\begin{displaymath}
\itexarray{ 
     El_P(C) &\to& * 
     \\ 
     \downarrow^{\mathrlap{\pi_P}} &\Downarrow& \downarrow^{\mathrlap{pt}} 
     \\ 
     C &\to& Set
  }
\end{displaymath}

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

\begin{displaymath}
\itexarray{ 
     El_P(C)^{op} &\overset{\pi_P^{op}}{\to}& C^{op} 
     \\ 
     \downarrow &\Downarrow& \downarrow^{\mathrlap{Y}} 
     \\ 
     * & \underset{P}{\to}& [C,Set]
  }
\end{displaymath}
$El_P(C)$ is also often written with [[end|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 $P(c)$.


\item It is the (strict) [[oplax colimit]] of the composite $C \xrightarrow{P} \mathbf{Set} \xrightarrow{disc} \mathbf{Cat}$; see [[Grothendieck construction]].



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

\begin{displaymath}
Explode(-) := El_F(-).
\end{displaymath}
This is intended to illustrate the concept that constructing a category of elements is like ``unpacking'' or ``exploding'' a category into its elements.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

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

\begin{theorem}
\label{ColimitPreserving}\hypertarget{ColimitPreserving}{}
The functor $el : \mathbf{Set}^{C} \to \mathbf{Cat}$ is [[cocontinuous functor|cocontinuous]].

\end{theorem}
\begin{proof}
As remarked above, $el$ is a strict [[weighted colimit]] [[2-colimit]], hence we have an isomorphism

\begin{displaymath}
el(P) \cong \int^{c\in C} J(c) \times disc(P(c))
\end{displaymath}
where the weight $J:C^{op} \to \mathbf{Cat}$ is the functor $c\mapsto c/C$, and $disc:\mathbf{Set}\hookrightarrow \mathbf{Cat}$ is the inclusion of the [[discrete categories]]. But since $disc$ (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.

\end{proof}
\begin{theorem}
\label{ColimitPreserving2}\hypertarget{ColimitPreserving2}{}
The functor $el\colon \mathbf{Set}^{C} \to \mathbf{Cat}$ has a right adjoint (which is maybe a more direct way to see that it is cocontinuous).

\end{theorem}
\begin{proof}
By a simple coend computation

\begin{displaymath}
\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}
\end{displaymath}
where $K(D)\colon c\mapsto [J c,D]_0$.

\end{proof}
Now for any $C$, the terminal object of $\mathbf{Set}^C$ 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 \emph{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 $\mathbf{Set}^C$ into $\mathbf{Cat}/C$.

Note that while the canonical projection $\operatorname{El}(F) \to \mathbf{C}$ is surjective on objects, it is not usually [[full functor|full]]. For example, let $\mathbf{B}\mathbb{N}$ be the one-object category which carries the monoid $(\mathbb{N}, +)$ as its endomorphism monoid, and let $F$ be the action of $(\mathbb{N}, +)$ on the set $\mathbb{N}$ by $n.m = m + n$. Then the image of any hom-set between $k, k'$ is a singleton subset of $\mathbb{N}$.

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 $0$ or $1$, while objects in the groupoid might have nontrivial automorphism groups.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{representable_presheaves}{}\subsubsection*{{Representable Presheaves}}\label{representable_presheaves}

Let $Y(C):\mathcal{C}^{op}\to Set$ be a [[representable functor|representable presheaf]] with $Y(C)(D)=Hom_{\mathcal{C}}(D,C)$. Consider the contravariant category of elements $\int_\mathcal{C} Y(C)$ . This has objects $(D_1,p_1)$ with $p_1\in Y(C)(D_1)$, hence $p_1$ is just an arrow $D_1\to C$ in $\mathcal{C}$. A map from $(D_1, p_1)$ to $(D_2, p_2)$ is just a map $u:D_1\to D_2$ such that $p_2\circ u =p_1$ but this is just a morphism from $p_1$ to $p_2$ in the [[overcategory|slice category]] $\mathcal{C}/C$. Accordingly we see that $\int_\mathcal{C} Y(C)\simeq \mathcal{C}/C$ .

This equivalence comes in handy when one wants to compute [[category of presheaves|slices of presheaf toposes]] over representable presheaves $Y(C)$ since $PSh(\int_\mathcal{C} P) \simeq PSh(\mathcal{C})/P$ in general for presheaves $P:\mathcal{C}^{op}\to Set$ , whence $PSh(\mathcal{C})/Y(C) \simeq PSh(\mathcal{C}/C)$ . An instructive example of this construction is spelled out in detail at [[hypergraph]].

\hypertarget{action_groupoid}{}\subsubsection*{{Action Groupoid}}\label{action_groupoid}

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

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

\begin{displaymath}
\itexarray{
{*} & \overset{x}{\to}  & X  \\
{}^1\downarrow && \downarrow^g\\
{*} & \underset{y}{\to} & X
}
\end{displaymath}
commutes, namely $g . x=\varrho(g)(x)=y$. We can also present the right adjoint to $el(-)$: one must consider the functor $J\colon \mathbf{B}G^{op}\to \mathbf{Cat}$, which represents $G$ in $\mathbf{Cat}$, and sends the unique object $*\in \mathbf{B}G$ to $*/\mathbf{B}G\cong G/\!/G$, the \emph{left} action groupoid of $G$. The functor $J$ sends $h\in G$ to an automorphism of $G/\!/G$, obtained multiplying \emph{on the right} $x\to g x$ to $x h\to x g h$.

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 $F^h=J(h)^*F=F \circ J(h) \colon g \mapsto F(g h)$.

\end{proof}
\hypertarget{category_of_simplices}{}\subsubsection*{{Category of simplices}}\label{category_of_simplices}

For a [[simplicial set]] regarded as a [[presheaf]] on the [[simplex category]], the corresponding category of elements is called its \emph{[[category of simplices]]}. See there for more.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Grothendieck construction]]


\item [[(∞,1)-Grothendieck construction]]


\item [[semi-direct product]]



\end{itemize}
\hypertarget{reference}{}\subsection*{{Reference}}\label{reference}

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

\begin{itemize}%
\item [[Michael Barr]], [[Charles Wells]], \emph{Category Theory for Computing Science} , Prentice Hall 1995$^3$. (\href{http://www.tac.mta.ca/tac/reprints/articles/22/tr22abs.html}{TAC reprints no.22 (2012)})

\end{itemize}
[[!redirects Exploding a Category]]



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