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\begin{document}

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\section*{indexed category}

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

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

[[!include category theory - contents]]

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,2)$-Topos theory}}\label{topos_theory}

[[!include (infinity,2)-topos 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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{self_indexing}{Self indexing}\dotfill \pageref*{self_indexing} \linebreak
\noindent\hyperlink{base_change}{Base change}\dotfill \pageref*{base_change} \linebreak
\noindent\hyperlink{indexed_category_of_a_functor}{Indexed category of a functor}\dotfill \pageref*{indexed_category_of_a_functor} \linebreak
\noindent\hyperlink{indexed_category_of_a_topos_over_a_base_topos}{Indexed category of a topos over a base topos}\dotfill \pageref*{indexed_category_of_a_topos_over_a_base_topos} \linebreak
\noindent\hyperlink{hyperdoctrine}{Hyperdoctrine}\dotfill \pageref*{hyperdoctrine} \linebreak
\noindent\hyperlink{indexed_monoidal_category}{Indexed monoidal category}\dotfill \pageref*{indexed_monoidal_category} \linebreak
\noindent\hyperlink{indexed_category}{Indexed $(\infty, 1)$-category}\dotfill \pageref*{indexed_category} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{extensions_of_adjunctions_to_indexed_categories}{Extensions of adjunctions to indexed categories}\dotfill \pageref*{extensions_of_adjunctions_to_indexed_categories} \linebreak
\noindent\hyperlink{WellPoweredness}{Well-powered indexed categories}\dotfill \pageref*{WellPoweredness} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

An \emph{indexed category} is a [[2-presheaf]].

When doing [[category theory]] relative to a [[base topos]] $\mathcal{S}$ (or other more general sort of [[category]]), the objects of $\mathcal{S}$ are thought of as replacements for [[sets]]. Since often in category theory we need to speak of ``a set-indexed family of objects'' of some category, we need a corresponding notion in ``category theory over $\mathcal{S}$.'' An \textbf{$\mathcal{S}$-indexed category} is a [[category]] $\mathbb{C}$ together with, for every [[object]] $X\in \mathcal{S}$, a notion of ``$X$-indexed family of objects of $\mathbb{C}$.''

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

Let $\mathcal{S}$ be a [[category]].

\begin{defn}
\label{}\hypertarget{}{}
An \textbf{$\mathcal{S}$-indexed category} $C$ is a [[pseudofunctor]]

\begin{displaymath}
\mathbb{C} : \mathcal{S}^{op}\to Cat
\end{displaymath}
from the [[opposite category]] of $\mathcal{S}$ to the [[2-category]] [[Cat]] of categories.

Under the [[Grothendieck construction]] [[equivalence of categories|equivalence]] this is equivalently a [[fibered category]]

\begin{displaymath}
\itexarray{
    \tilde \mathbb{C} \\ \downarrow \\ \mathcal{S}
  }
\end{displaymath}
over $\mathcal{S}$.

Similarly, an $\mathcal{S}$-[[indexed functor]] $\mathbb{C} \to \mathbb{D}$ is a [[pseudonatural transformation]] of pseudofunctors, and an indexed natural transformation is a [[modification]].

This defines the [[2-category]] $\mathcal{S} IndCat \coloneqq [\mathcal{S}^{op}, Cat]$ of $\mathcal{S}$-indexed categories.

\end{defn}
This appears for instance as (\hyperlink{Johnstone}{Johnstone, def. B1.2.1}).

One may also call $\mathbb{C}$ a [[prestack]] in categories over $\mathcal{S}$.

Traditionally one writes the image of an object $X \in \mathcal{S}$ under $\mathbb{C}$ as $\mathbb{C}^X$ and calls it \emph{the category of $X$-indexed families of objects of $\mathbb{C}$}. Similarly, one writes the image of a [[morphism]] $u\colon X\to Y$ as $u^*\colon \mathbb{C}^Y\to \mathbb{C}^X$.

If $\mathcal{S}$ has a [[terminal object]] $*$ we think of $\mathbb{C}^*$ as the \textbf{underlying ordinary category} of the $\mathcal{S}$-indexed category $\mathbb{C}$. Part of the theory of indexed categories is about when and how to extend structures on $\mathbb{C}^*$ to all of $\mathbb{C}$.

A morphism of $S$-indexed categories is an [[indexed functor]].

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

\hypertarget{self_indexing}{}\subsubsection*{{Self indexing}}\label{self_indexing}

\begin{example}
\label{CanonicalSelfIndexing}\hypertarget{CanonicalSelfIndexing}{}
\textbf{(canonical self-indexing)}

If $\mathcal{S}$ has [[pullbacks]], then its [[codomain fibration]] is an $\mathcal{S}$-indexed category denoted $\mathbb{S}$.

This assigns to an object $I$ the corresponding [[over-category]]

\begin{displaymath}
\mathbb{S}^I \coloneqq \mathcal{S}/I
\end{displaymath}
and to a morphism $f : I \to J$ the functor $f^*$ that sends every $s \to I$ to its [[pullback]] $f^*$ along $f$.

\end{example}
This indexed category represents $\mathcal{S}$ itself (or rather its [[codomain fibration]]) in the world of $\mathcal{S}$-indexed categories.

\hypertarget{base_change}{}\subsubsection*{{Base change}}\label{base_change}

\begin{example}
\label{BaseChange}\hypertarget{BaseChange}{}
\textbf{(change of base)}

If $F\colon \mathcal{S}\to \mathcal{T}$ is a [[functor]] and $\mathbb{C}$ is a $\mathcal{T}$-indexed category, then we have an $\mathcal{S}$-indexed category $F^*\mathbb{C}$ defined by

\begin{itemize}%
\item $(F^*\mathbb{C})^I = \mathbb{C}^{F(I)}$ for every object $I \in \mathcal{S}$;


\item and $x^* = F(x)^*$ for every morphism $x : I \to J$ in $\mathcal{S}$.



\end{itemize}
\end{example}
\hypertarget{indexed_category_of_a_functor}{}\subsubsection*{{Indexed category of a functor}}\label{indexed_category_of_a_functor}

Combining these previous examples we get

\begin{example}
\label{CartesianFunctorIndexing}\hypertarget{CartesianFunctorIndexing}{}
For $F : \mathcal{S} \to \mathcal{C}$ a functor and $\mathcal{C}$ a [[finitely complete category]], there is the $\mathcal{S}$-indexed category $F^* \mathbb{C}$ given by

\begin{itemize}%
\item $(F^* \mathbb{C})^I = \mathcal{C}/F(I)$.

\end{itemize}
If the functor $F$ preserves [[pullback]]s then this induces a morphism $\mathbb{S} \to F^* \mathbb{C}$ of $\mathcal{S}$-indexed categories.

\end{example}
\hypertarget{indexed_category_of_a_topos_over_a_base_topos}{}\subsubsection*{{Indexed category of a topos over a base topos}}\label{indexed_category_of_a_topos_over_a_base_topos}

This situation frequently arises when $\mathcal{S}$ and $\mathcal{C}$ are [[toposes]] and $F \coloneqq f^*$ is the [[inverse image]] part of a [[geometric morphism]].

\begin{displaymath}
f 
  : 
  \mathcal{C}
   \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}
  \mathcal{S}
 \,.
\end{displaymath}
In this way, if $\mathcal{S}$ is a topos, then to be thought of as a \emph{[[base topos]]}, then any topos \emph{over} $\mathcal{S}$ (i.e. an object of the [[slice 2-category]] [[Topos]]$/S$) gives rise to a topos \emph{relative to} $\mathcal{S}$, i.e. a ``topos object'' in the 2-category of $\mathcal{S}$-indexed categories, and this operation can be shown to be fully faithful.

See [[base topos]] for more on this.

Also, via this indexed category, $f$ exhibits $\mathcal{C}$ as a [[2-sheaf]] (see there) over $\mathcal{C}$, with respect to the [[canonical topology]].

\hypertarget{hyperdoctrine}{}\subsubsection*{{Hyperdoctrine}}\label{hyperdoctrine}

\begin{itemize}%
\item [[hyperdoctrine]]

\end{itemize}
\hypertarget{indexed_monoidal_category}{}\subsubsection*{{Indexed monoidal category}}\label{indexed_monoidal_category}

See also \emph{[[indexed monoidal category]]}, \emph{[[indexed closed monoidal category]]} and \emph{[[dependent linear type theory]]}.

\hypertarget{indexed_category}{}\subsubsection*{{Indexed $(\infty, 1)$-category}}\label{indexed_category}

See \emph{[[indexed (infinity, 1)-category]]}.

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

\hypertarget{extensions_of_adjunctions_to_indexed_categories}{}\subsubsection*{{Extensions of adjunctions to indexed categories}}\label{extensions_of_adjunctions_to_indexed_categories}

\begin{prop}
\label{}\hypertarget{}{}
Let

\begin{displaymath}
(L \dashv R) : 
   \mathcal{C} \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} \mathcal{S}
\end{displaymath}
be a pair of [[adjoint functor]]s between [[finitely complete categories]]. Then $R$ extends to an $\mathcal{S}$-indexed functor

\begin{displaymath}
\mathbb{R} : \mathbb{C} \to \mathbb{S}
\end{displaymath}
where $\mathbb{S}$ is the self-indexing of $\mathcal{S}$ from example \ref{CanonicalSelfIndexing} and $\mathbb{C}$ is the base change indexing of $\mathcal{C}$ from example \ref{CartesianFunctorIndexing}.

By the general properties of adjunctions on [[overcategories]] (see there) we get for each $I \in \mathcal{S}$ an adjunction

\begin{displaymath}
(L/I \dashv R/I) : \mathbb{C}^I = \mathcal{C}/R(I) \to \mathcal{S}/I = \mathbb{S}^I
  \,.
\end{displaymath}
Here $\mathbb{R} : I \mapsto R/I$ is always a $\mathcal{S}$-indexed functor $\mathbb{C} \to \mathbb{S}$, and $\mathbb{L} : I \mapsto L/I$ is if $L$ preserves [[pullback]]s (by example \ref{CartesianFunctorIndexing}). If so, we have an $\mathcal{S}$-indexed adjunction

\begin{displaymath}
(\mathbb{L} \dashv \mathbb{R}) : 
  \mathbb{C} \to \mathbb{S}
\end{displaymath}
\end{prop}
This appears as (\hyperlink{Johnstone}{Johnstone, lemma B1.2.3}).

\begin{proof}
(\ldots{})

\end{proof}
\hypertarget{WellPoweredness}{}\subsubsection*{{Well-powered indexed categories}}\label{WellPoweredness}

\begin{defn}
\label{}\hypertarget{}{}
An $\mathcal{S}$-indexed category $\mathbb{C}$ is called \textbf{well-powered} if the [[fibered category]] $\tilde \mathbb{C} \to \mathcal{S}$ corresponding to it under the [[Grothendieck construction]] has the property that the [[forgetful functor]]

\begin{displaymath}
U : Q(2, \tilde \mathbb{C})
   \to
  Rect(*,\tilde \mathbb{C})
\end{displaymath}
has a [[right adjoint]], where $Q(2,\tilde \mathbb{C})$ is the [[full subcategory]] of $Rect(2, \tilde \mathbb{C})$ on vertical [[monomorphism]]s.

\end{defn}
This appears as (\hyperlink{Johnstone}{Johnstone, example. B1.3.14}).

\begin{prop}
\label{}\hypertarget{}{}
Let $(L \dashv R) : \mathcal{C} \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} \mathcal{S}$ be a pair of [[adjoint functor]]s such that $L$ preserves pullbacks. Then the $\mathcal{S}$-indexed category $\mathbb{C}$ is well powered if $\mathbb{S}$ is.

\end{prop}
This is (\hyperlink{Johnstone}{Johnstone, prop. B1.3.17}).

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

\begin{itemize}%
\item [[indexed functor]]


\item [[indexed topos]]


\item [[hyperdoctrine]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Section B1.2 in

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]}

\end{itemize}
[[!redirects indexed categories]]



\end{document}