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\begin{document}

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\section*{base change}

\begin{quote}%
This entry is about base change of [[slice categories]]. For base change in [[enriched category theory]] see at [[change of enriching category]].


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{limits_and_colimits}{}\paragraph*{{Limits and colimits}}\label{limits_and_colimits}

[[!include infinity-limits - contents]]

\hypertarget{topos_theory}{}\paragraph*{{Topos theory}}\label{topos_theory}

[[!include 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{pullback}{Pullback}\dotfill \pageref*{pullback} \linebreak
\noindent\hyperlink{in_a_fibered_category}{In a fibered category}\dotfill \pageref*{in_a_fibered_category} \linebreak
\noindent\hyperlink{GeometricMorphism}{Base change geometric morphisms}\dotfill \pageref*{GeometricMorphism} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{AlongDeloopingsOfGroupHomomorphisms}{Along $\mathbf{B}H \to \mathbf{B}G$}\dotfill \pageref*{AlongDeloopingsOfGroupHomomorphisms} \linebreak
\noindent\hyperlink{AlongPointInclusionIntoBG}{Along $\ast \to \mathbf{B}G$}\dotfill \pageref*{AlongPointInclusionIntoBG} \linebreak
\noindent\hyperlink{along__3}{Along $V/G \to \mathbf{B}G$}\dotfill \pageref*{along__3} \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}

For $f : X \to Y$ a [[morphism]] in a [[category]] $C$ with [[pullbacks]], there is an induced [[functor]]

\begin{displaymath}
f^* : C/Y \to C/X
\end{displaymath}
of [[over-categories]]. This is the \emph{base change} morphism. If $C$ is a [[topos]], then this refines to an [[essential geometric morphism]]

\begin{displaymath}
(f_! \dashv f^* \dashv f_*) : C/X \to C/Y
  \,.
\end{displaymath}
The [[duality|dual]] concept is [[cobase change]].

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

\hypertarget{pullback}{}\subsubsection*{{Pullback}}\label{pullback}

For $f : X \to Y$ a [[morphism]] in a [[category]] $C$ with [[pullback]]s, there is an induced [[functor]]

\begin{displaymath}
f^* : C/Y \to C/X
\end{displaymath}
of [[over-categories]]. It is on objects given by [[pullback]]/[[fiber product]] along $f$

\begin{displaymath}
(p : K \to Y)
  \mapsto
  \left(
  \itexarray{
    X \times_Y K &\to & K
    \\
    {}^{\mathllap{p^*}}\downarrow && \downarrow
    \\
    X &\to& Y 
  }
  \right)
  \,.
\end{displaymath}
\hypertarget{in_a_fibered_category}{}\subsubsection*{{In a fibered category}}\label{in_a_fibered_category}

The concept of base change generalises from this case to other [[fibered category|fibred categories]].

\hypertarget{GeometricMorphism}{}\subsubsection*{{Base change geometric morphisms}}\label{GeometricMorphism}

\begin{prop}
\label{BaseChangeIsEssentialGeometricMorphism}\hypertarget{BaseChangeIsEssentialGeometricMorphism}{}
For $\mathbf{H}$ a [[topos]] (or [[(∞,1)-topos]], etc.) $f : X \to Y$ a [[morphism]] in $\mathbf{H}$, then base change induces an [[essential geometric morphism]] between over-toposes/[[over-(∞,1)-topos]]es

\begin{displaymath}
(\sum_f \dashv f^* \dashv \prod_f) : \mathbf{H}/X 
   \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}}
 \mathbf{H}/Y
\end{displaymath}
where $f_!$ is given by postcomposition with $f$ and $f^*$ by [[pullback]] along $f$.

\end{prop}
\begin{proof}
That we have [[adjoint functor]]s/[[adjoint (∞,1)-functor]]s $(f_! \dashv f^*)$ follows directly from the universal property of the pullback. The fact that $f^*$ has a further [[right adjoint]] is due to the fact that it preserves all small [[colimit]]s/[[(∞,1)-colimit]]s by the fact that in a topos we have [[universal colimits]] and then by the [[adjoint functor theorem]]/[[adjoint (∞,1)-functor theorem]].

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
The ([[comonad|co-]])[[monads]] induced by the [[adjoint triple]] in prop. \ref{BaseChangeIsEssentialGeometricMorphism} have special names in some contexts:

\begin{itemize}%
\item $f_\ast f^\ast$ is also called the [[function monad]] (or ``[[reader monad]]'', see at \emph{[[monad (in computer science)]]}).


\item $f_! f^\ast$ is also called the ``[[writer comonad]]'' (in computer science)


\item in [[modal type theory]] $f^\ast f_\ast$ is \emph{[[necessity]]} while $f^\ast f_!$ is \emph{[[possibility]]}.



\end{itemize}
\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
Here $f^\ast$ is a [[cartesian closed functor]], hence base change of toposes constitutes a cartesian [[Wirthmüller context]].

\end{prop}
See at \emph{[[cartesian closed functor]]} for the proof.

\begin{prop}
\label{}\hypertarget{}{}
$f^*$ is a [[logical functor]]. Hence $(f^* \dashv f_*)$ is also an [[atomic geometric morphism]].

\end{prop}
This appears for instance as (\hyperlink{MacLaneMoerdijk}{MacLaneMoerdijk, theorem IV.7.2}).

\begin{proof}
By prop. \ref{BaseChangeIsEssentialGeometricMorphism} $f^*$ is a [[right adjoint]] and hence preserves all [[limit]]s, in particular [[finite limit]]s.

Notice that the [[subobject classifier]] of an [[over topos]] $\mathbf{H}/X$ is $(p_2 : \Omega_{\mathbf{H}} \times X \to X)$. This [[product]] is preserved by the [[pullback]] by which $f^*$ acts, hence $f^*$ preserves the subobject classifier.

To show that $f^*$ is logical it therefore remains to show that it also preserves [[exponential object]]s. (\ldots{})

\end{proof}
\begin{defn}
\label{}\hypertarget{}{}
A (necessarily essential and atomic) geometric morphism of the form $(f^* \dashv \prod_f)$ is called the \textbf{base change geometric morphism} along $f$.

The [[right adjoint]] $f_* = \prod_f$ is also called the [[dependent product]] relative to $f$.

The [[left adjoint]] $f_! = \sum_f$ is also called the [[dependent sum]] relative to $f$.

In the case $Y = *$ is the [[terminal object]], the base change geometric morphism is also called an \textbf{[[etale geometric morphism]]}. See there for more details

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

\begin{prop}
\label{}\hypertarget{}{}
If $\mathcal{C}$ is a [[locally cartesian closed category]] then for every morphism $f \colon X \to Y$ in $f$ the [[inverse image]] $f^* \colon \mathcal{C}_{/Y} \to \mathcal{C}_{/X}$ of the base change is a [[cartesian closed functor]].

\end{prop}
See at \emph{\href{cartesian%20closed%20functor#Examples}{cartesian closed functor -- Examples}} for a proof.

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

\hypertarget{AlongDeloopingsOfGroupHomomorphisms}{}\subsubsection*{{Along $\mathbf{B}H \to \mathbf{B}G$}}\label{AlongDeloopingsOfGroupHomomorphisms}

For $\mathbf{H}$ an [[(∞,1)-topos]] and $G$ an group object in $\mathbf{H}$ (an [[∞-group]]), then the [[slice (∞,1)-topos]] over its [[delooping]] may be identified with the [[(∞,1)-category]] of $G$-[[∞-actions]] (see there for more):

\begin{displaymath}
Act_G(\mathbf{H}) \simeq \mathbf{H}_{/\mathbf{B}G}
  \,.
\end{displaymath}
Under this identification, then left and right base change long a morphism of the form $\mathbf{B}H \to \mathbf{B}G$ (corresponding to an [[∞-group]] homomorphism $H \to G$) corresponds to forming [[induced representations]] and [[coinduced representations]], respectively.

\hypertarget{AlongPointInclusionIntoBG}{}\subsubsection*{{Along $\ast \to \mathbf{B}G$}}\label{AlongPointInclusionIntoBG}

As the special case of the \hyperlink{AlongDeloopingsOfGroupHomomorphisms}{above} for $H = 1$ the trivial group we obtain the following:

\begin{prop}
\label{CyclicLoopSpace}\hypertarget{CyclicLoopSpace}{}
Let $\mathbf{H}$ be any [[(∞,1)-topos]] and let $G$ be a group object in $\mathbf{H}$ (an [[∞-group]]). Then the base change along the canonical point inclusion

\begin{displaymath}
i \;\colon\; \ast \to \mathbf{B}G
\end{displaymath}
into the [[delooping]] of $G$ takes the following form:

There is a pair of [[adjoint ∞-functors]] of the form

\begin{displaymath}
\mathbf{H}
    \underoverset
      { \underset{i_\ast \simeq [G,-]/G}{\longrightarrow}}
      { \overset{i^\ast \simeq hofib}{\longleftarrow}}
      {\bot}
  \mathbf{H}_{/\mathbf{B}G}
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $hofib$ denotes the operation of taking the [[homotopy fiber]] of a map to $\mathbf{B}G$ over the canonical basepoint;


\item $[G,-]$ denotes the [[internal hom]] in $\mathbf{H}$;


\item $[G,-]/G$ denotes the [[homotopy quotient]] by the [[conjugation action|conjugation]] [[∞-action]] for $G$ equipped with its canonical [[∞-action]] by left multiplication and the argument regarded as equipped with its trivial $G$-$\infty$-action

(for $G = S^1$ then this is the [[cyclic loop space]] construction).



\end{itemize}
Hence for

\begin{itemize}%
\item $\hat X \to X$ a $G$-[[principal ∞-bundle]]


\item $A$ a [[coefficient]] object, such as for some [[differential cohomology|differential]] [[generalized cohomology theory]]



\end{itemize}
then there is a [[natural equivalence]]

\begin{displaymath}
\underset{
    \text{original} \atop \text{fluxes}
  }{
  \underbrace{
    \mathbf{H}(\hat X\;,\; A)
  }
  }
    \;\;
    \underoverset
      {\underset{oxidation}{\longleftarrow}}
      {\overset{reduction}{\longrightarrow}}
      {\simeq}
    \;\;
  \underset{
     \text{doubly} \atop { \text{dimensionally reduced} \atop \text{fluxes} }
  }{
  \underbrace{
    \mathbf{H}(X \;,\; [G,A]/G)
  }
  }
\end{displaymath}
given by

\begin{displaymath}
\left(
     \hat X \longrightarrow A
  \right)
  \;\;\;
    \leftrightarrow
  \;\;\;
  \left(
    \itexarray{
       X && \longrightarrow && [G,A]/G
       \\
       & \searrow && \swarrow
       \\
       && \mathbf{B}G
    }
  \right)
\end{displaymath}
\end{prop}
\begin{proof}
The statement that $i^\ast \simeq hofib$ follows immediately by the definitions. What we need to see is that the [[dependent product]] along $i$ is given as claimed.

To that end, first observe that the [[conjugation action]] on $[G,X]$ is the [[internal hom]] in the [[(∞,1)-category]] of $G$-[[∞-actions]] $Act_G(\mathbf{H})$. Under the [[equivalence of (∞,1)-categories]]

\begin{displaymath}
Act_G(\mathbf{H}) \simeq \mathbf{H}_{/\mathbf{B}G}
\end{displaymath}
(from \href{https://ncatlab.org/schreiber/show/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications}{NSS 12}) then $G$ with its canonical [[∞-action]] is $(\ast \to \mathbf{B}G)$ and $X$ with the trivial action is $(X \times \mathbf{B}G \to \mathbf{B}G)$.

Hence

\begin{displaymath}
[G,X]/G
    \simeq
  [\ast, X \times \mathbf{B}G]_{\mathbf{B}G}
  \;\;\;\;\;
  \in
   \mathbf{H}_{/\mathbf{B}G}
  \,.
\end{displaymath}
So far this is the very definition of what $[G,X]/G \in \mathbf{H}_{/\mathbf{B}G}$ is to mean in the first place.

But now since the [[slice (∞,1)-topos]] $\mathbf{H}_{/\mathbf{B}G}$ is itself [[cartesian closed (infinity,1)-category|cartesian closed]], via

\begin{displaymath}
E \times_{\mathbf{B}G}(-)
  \;\;\;
   \dashv
  \;\;\;
  [E,-]_{\mathbf{B}G}
\end{displaymath}
it is immediate that there is the following sequence of [[natural equivalences]]

\begin{displaymath}
\begin{aligned}
   \mathbf{H}_{/\mathbf{B}G}(Y, [G,X]/G)
   & \simeq
   \mathbf{H}_{/\mathbf{B}G}(Y, [\ast, X \times \mathbf{B}G]_{\mathbf{B}G})
   \\
   & \simeq
   \mathbf{H}_{/\mathbf{B}G}(
     Y \times_{\mathbf{B}G} \ast,
     \underset{p^\ast X}{\underbrace{X \times \mathbf{B}G }}
  )
  \\
  & \simeq
  \mathbf{H}(
     \underset{hofib(Y)}{\underbrace{p_!(Y \times_{\mathbf{B}G} \ast)}},
     X
  )
  \\
  & \simeq
  \mathbf{H}(hofib(Y),X)
  \end{aligned}
\end{displaymath}
Here $p \colon \mathbf{B}G \to \ast$ denotes the terminal morphism and $p_! \dashv p^\ast$ denotes the [[base change]] along it.

\end{proof}
See also at \emph{[[double dimensional reduction]]} for more on this.

\hypertarget{along__3}{}\subsubsection*{{Along $V/G \to \mathbf{B}G$}}\label{along__3}

More generally:

\begin{prop}
\label{RightBaseChangeAlongUniversalFiberBundleProjection}\hypertarget{RightBaseChangeAlongUniversalFiberBundleProjection}{}
Let $\mathbf{H}$ be an [[(∞,1)-topos]] and $G \in Grp(\mathbf{H})$ an [[∞-group]].

Let moreover $V \in \mathbf{H}$ be an object equipped with a $G$-[[∞-action]] $\rho$, equivalently (by the discussion there) a [[homotopy fiber sequence]] of the form

\begin{displaymath}
\itexarray{
    V
    \\
    \downarrow
    \\
    V/G & \overset{p_\rho}{\longrightarrow}& \mathbf{B}G
  }
\end{displaymath}
Then

\begin{enumerate}%
\item pullback along $p_\rho$ is the operation that assigns to a morphism $c \colon X \to \mathbf{B}G$ the $V$-[[fiber ∞-bundle]] which is [[associated ∞-bundle|associated]] via $\rho$ to the $G$-[[principal ∞-bundle]] $P_c$ classified by $c$:

\begin{displaymath}
(p_\rho)^\ast \;\colon\; c \mapsto P_c \times_G V
\end{displaymath}

\item the right base change along $p_\rho$ is given on objects of the form $X \times (V/G)$ by

\begin{displaymath}
(p_\rho)_\ast \;\colon\; X \times (V/G) \;\mapsto\;  [V,X]/G
\end{displaymath}


\end{enumerate}
\end{prop}
\begin{proof}
The first statement is \href{https://ncatlab.org/schreiber/show/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications}{NSS 12, prop. 4.6}.

The second statement follows as in the proof of prop. \ref{CyclicLoopSpace}: Let

\begin{displaymath}
\left(
    \itexarray{
      Y
      \\
      \downarrow^{\mathrlap{c}}
      \\
      \mathbf{B}G
    }
  \right)
  \;\in\;
  \mathbf{H}_{/\mathbf{B}G}
\end{displaymath}
be any object, then there is the following sequence of [[natural equivalences]]

\begin{displaymath}
\begin{aligned}
   \mathbf{H}_{/\mathbf{B}G}(Y, [V,X]/G)
   & \simeq
   \mathbf{H}_{/\mathbf{B}G}(Y, [V/G, X \times \mathbf{B}G]_{\mathbf{B}G})
   \\
   & \simeq
   \mathbf{H}_{/\mathbf{B}G}(
     Y \times_{\mathbf{B}G} (V/G),
     \underset{p^\ast X}{\underbrace{X \times \mathbf{B}G }}
  )
  \\
  & \simeq
  \mathbf{H}_{/\mathbf{B}G}
  (
    (p_\rho)_!( P_c \times_G (V/G) ),
    p^\ast X
  )
  \\
  & \simeq
     \mathbf{H}_{/(V/G)}
     (
       P_c \times_G V, (p_\rho)^\ast p^\ast X
     )
  \\
  & 
  \simeq
    \mathbf{H}_{/(V/G)}(P_c \times_G V, X \times (V/G))
  \end{aligned}
\end{displaymath}
where again

\begin{displaymath}
p \colon \mathbf{B}G \to \ast
  \,.
\end{displaymath}
\end{proof}
\begin{example}
\label{SymmetricPowers}\hypertarget{SymmetricPowers}{}
\textbf{(symmetric powers)}

Let

\begin{displaymath}
G = \Sigma(n) \in Grp(Set) \hookrightarrow Grp(\infty Grpd) \overset{LConst}{\longrightarrow} \mathbf{H}
\end{displaymath}
be the [[symmetric group]] on $n$ elements, and

\begin{displaymath}
V = \{1, \cdots, n\} \in Set \hookrightarrow \infty Grpd \overset{LConst}{\longrightarrow} \mathbf{H}
\end{displaymath}
the $n$-element [[set]] ([[h-set]]) equipped with the canonical $\Sigma(n)$-[[action]]. Then prop. \ref{RightBaseChangeAlongUniversalFiberBundleProjection} says that right base change of any $p_\rho^\ast p^\ast X$ along

\begin{displaymath}
\{1, \cdots, n\}/\Sigma(n)
    \longrightarrow
  \mathbf{B}\Sigma(n)
\end{displaymath}
is equivalently the $n$th [[symmetric power]] of $X$

\begin{displaymath}
[\{1,\cdots, n\},X]/\Sigma(n)
  \simeq 
  (X^n)/\Sigma(n)
  \,.
\end{displaymath}
\end{example}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[pullback]], [[fiber product]],


\item [[lax pullback]], [[comma object]],


\item [[(∞,1)-pullback]], [[homotopy pullback]]


\item \textbf{base change}

\begin{itemize}%
\item [[dependent sum]], [[dependent product]]


\item [[dependent sum type]], [[dependent product type]]


\item [[necessity]], [[possibility]], [[reader monad]], [[writer comonad]]



\end{itemize}

\item [[proper base change theorem]]


\item Base change geometric morphisms may be interpreted in terms of [[fiber integration]]. See [[integral transforms on sheaves]] for more on this.


\item [[change of enriching category]]



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

A general discussion that applies (also) to [[enriched categories]] and [[internal categories]] is in

\begin{itemize}%
\item [[Dominic Verity]], \emph{Enriched categories, internal categories and change of base} Ph.D. thesis, Cambridge University (1992), reprinted as Reprints in Theory and Applications of Categories, No. 20 (2011) pp 1-266 (\href{http://www.tac.mta.ca/tac/reprints/articles/20/tr20abs.html}{TAC})

\end{itemize}
Discussion in the context of [[topos theory]] is around example A.4.1.2 of

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

\end{itemize}
and around theorem IV.7.2 in

\begin{itemize}%
\item [[Saunders MacLane]], [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]}

\end{itemize}
Discussion in the context of [[(infinity,1)-topos theory]] is in section 6.3.5 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

\end{itemize}
See also

\begin{itemize}%
\item A. Carboni, G. Kelly, R. Wood, \emph{A 2-categorical approach to change of base and geometric morphisms I} (\href{http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1991__32_1/CTGDC_1991__32_1_47_0/CTGDC_1991__32_1_47_0.pdf}{numdam})

\end{itemize}
[[!redirects change of base]]

[[!redirects base change geometric morphism]] [[!redirects base change geometric morphisms]]



\end{document}