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\section*{Beck-Chevalley condition}

\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{left_and_right_beckchevalley_condition}{Left and right Beck--Chevalley condition}\dotfill \pageref*{left_and_right_beckchevalley_condition} \linebreak
\noindent\hyperlink{dual_beckchevalley_condition}{Dual Beck--Chevalley condition}\dotfill \pageref*{dual_beckchevalley_condition} \linebreak
\noindent\hyperlink{for_bifibrations}{For bifibrations}\dotfill \pageref*{for_bifibrations} \linebreak
\noindent\hyperlink{Local}{``Local'' Beck--Chevalley condition}\dotfill \pageref*{Local} \linebreak
\noindent\hyperlink{InTypeTheory}{In logic / type theory}\dotfill \pageref*{InTypeTheory} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{various}{Various}\dotfill \pageref*{various} \linebreak
\noindent\hyperlink{PullbacksOfOpfibrations}{For categories of presheaves}\dotfill \pageref*{PullbacksOfOpfibrations} \linebreak
\noindent\hyperlink{proper_base_change_in_tale_cohomology}{Proper base change in \'e{}tale cohomology}\dotfill \pageref*{proper_base_change_in_tale_cohomology} \linebreak
\noindent\hyperlink{grothendieck_six_operations}{Grothendieck six operations}\dotfill \pageref*{grothendieck_six_operations} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \textbf{Beck--Chevalley condition}, also sometimes called just the \emph{Beck condition} or the \emph{Chevalley condition}, is a ``commutation of [[adjoint functor|adjoint]]s'' property that holds in many ``[[base change|change of base]]'' situations.

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

Suppose given a [[commutative square]] (up to [[isomorphism]]) of [[functors]]:

\begin{displaymath}
\itexarray{ & \overset{f^*}{\to} & \\
  ^{g^*}\downarrow && \downarrow^{k^*}\\
  & \underset{h^*}{\to} & }
\end{displaymath}
in which $f^*$ and $h^*$ have [[left adjoint]]s $f_!$ and $h_!$, respectively. (The classical example is a [[Wirthmüller context]].) Then the [[natural isomorphism]] that makes the square commute

\begin{displaymath}
k^* f^* \to h^* g^*
\end{displaymath}
has a [[mate]]

\begin{displaymath}
h_! k^* \to g^* f_!
\end{displaymath}
defined as the composite

\begin{displaymath}
h_! k^* \overset{\eta}{\to} h_! k^* f^* f_! \overset{\cong}{\to} h_! h^* g^* f_! \overset{\epsilon}{\to} g^* f_!   
  \,.
\end{displaymath}
We say the original square satisfies the \textbf{Beck--Chevalley condition} if this mate is an [[isomorphism]].

More generally, it is clear that for this to make sense, we only need a transformation $k^* f^* \to h^* g^*$; it doesn't need to be an isomorphism. We also use the term \emph{Beck--Chevalley condition} in this case,

\hypertarget{left_and_right_beckchevalley_condition}{}\subsubsection*{{Left and right Beck--Chevalley condition}}\label{left_and_right_beckchevalley_condition}

Of course, if $g^*$ and $k^*$ also have [[left adjoints]], there is also a Beck--Chevalley condition stating that the corresponding mate $k_! h^* \to f^* g_!$ is an isomorphism, and this is not equivalent in general. Context is usually sufficient to disambiguate, although some people speak of the ``left'' and ``right'' Beck--Chevalley conditions.

Note that if $k^* f^* \to h^* g^*$ is not an isomorphism, then there is only one possible Beck-Chevalley condition.

\hypertarget{dual_beckchevalley_condition}{}\subsubsection*{{Dual Beck--Chevalley condition}}\label{dual_beckchevalley_condition}

If $f^*$ and $h^*$ have \emph{right} adjoints $f_*$ and $h_*$, there is also a dual Beck--Chevalley condition saying that the mate $g^* f_* \to h_* k^*$ is an isomorphism. By general nonsense, if $f^*$ and $h^*$ have right adjoints and $g^*$ and $k^*$ have left adjoints, then $g^* f_* \to h_* k^*$ is an isomorphism if and only if $k_! h^* \to f^* g_!$ is.

\hypertarget{for_bifibrations}{}\subsubsection*{{For bifibrations}}\label{for_bifibrations}

Originally, the Beck-Chevalley condition was introduced in (\hyperlink{BenabouRoubaud}{B\'e{}nabou-Roubaud, 1970}) for [[bifibrations]] over a base category with pullbacks. In \emph{loc.cit.} they call this condition \textbf{Chevalley condition} because he introduced it in his 1964 seminar.

A [[bifibration]] $\mathbf{X} \to \mathbf{B}$ where $\mathbf{B}$ has pullbacks satisfies the \textbf{Chevalley condition} iff for every commuting square

\begin{displaymath}
\itexarray{ & \overset{\psi^\prime}{\rightarrow} & \\
  \downarrow^{\varphi^\prime} && \downarrow^{\varphi}\\
  & \underset{\psi}{\rightarrow} & }
\end{displaymath}
in $\mathbf{X}$ over a pullback square in the base $\mathbf{B}$ where $\varphi$ is [[cartesian morphism|cartesian]] and $\psi$ is cocartesian it holds that $\varphi^\prime$ is cartesian iff $\psi^\prime$ is cocartesian. Actually, it suffices to postulate one direction because the other one follows. The nice thing about this formulation is that there is no mention of ``canonical'' morphisms and no mention of [[cleavages]].

A fibration $P$ has products satisfying the Chevalley condition iff the opposite fibration $P^{op}$ is a bifibration satisfying the Chevalley condition in the above sense.

According to the [[Benabou–Roubaud theorem]], the Chevalley condition is crucial for establishing the connection between the descent in the sense of fibered categories and the [[monadic descent]].

\hypertarget{Local}{}\subsubsection*{{``Local'' Beck--Chevalley condition}}\label{Local}

Suppose that $f^*$ and $h^*$ do not have entire left adjoints, but that for a particular object $x$ the left adjoint $f_!(x)$ exists. This means that we have an object ``$f_! x$'' and a morphism $\eta_x\colon x \to f^* f_! x$ which is initial in the [[comma category]] $(x / f^*)$. Then we have $k^*(\eta) \colon k^* x \to k^* f^* f_! x \to h^* g^* f_! x$, and we say that the square satisfies the \emph{local Beck-Chevalley condition at $x$} if $k^*(\eta)$ is initial in the comma category $(k^* x / h^*)$, and hence exhibits $g^* f_! x$ as ``$h_! k^* x$'' (although we have not assumed that the entire functor $h_!$ exists).

If the functors $f_!$ and $h_!$ do exist, then the square satisfies the (global) Beck-Chevalley condition as defined above if and only if it satisfies the local one defined here at every object.

\hypertarget{InTypeTheory}{}\subsection*{{In logic / type theory}}\label{InTypeTheory}

If the functors in the formulation of the Beck-Chevalley condition are [[base change]] functors in the [[categorical semantics]] of some [[dependent type theory]] (or just of a [[hyperdoctrine]]) then the BC condition is equivalently stated in terms of logic as follows.

A [[commuting diagram]]

\begin{displaymath}
\itexarray{
    D &\stackrel{h}{\to}& C 
    \\
    {}^{\mathllap{k}}\downarrow && \downarrow^{\mathrlap{g}}
    \\
    A &\stackrel{f}{\to}& B
  }
\end{displaymath}
is interpreted as a morphism of [[contexts]]. The [[pullback]] (of [[slice categories]] or of fibers in a [[hyperdoctrine]]) $h^*$ and $f^*$ is interpreted as the [[substitution]] of [[variables]] in these contexts. And the [[left adjoint]] $\sum_k \coloneqq k_!$ and $\sum_q \coloneqq g_!$, the [[dependent sum]] is interpreted (up to [[truncated|(-1)-truncation]], possibly) as [[existential quantifier|existential quantification]].

In terms of this the Beck-Chevalley condition says that if the above diagram is a [[pullback]], then \textbf{substitution commutes with existential quantification}

\begin{displaymath}
\sum_k h^* \phi \stackrel{\simeq}{\to} f^* \sum_g \phi
  \,.
\end{displaymath}
\begin{example}
\label{}\hypertarget{}{}
Consider the diagram of [[contexts]]

\begin{displaymath}
\itexarray{
    [\Gamma, x : X] &\stackrel{}{\to}& [\Gamma, x : X, y : Y]
    \\
    \downarrow && \downarrow
    \\
    \Gamma &\to& [\Gamma, y : Y]
  }
  \;\;\;
  \simeq
  \;\;\;
  \itexarray{
    \Gamma \times X &\stackrel{(p_1,p_2,t)}{\to}& \Gamma \times X \times Y
    \\
    {}^{\mathllap{p_1}}\downarrow && \downarrow^{\mathrlap{(p_1,p_3)}}
    \\
    \Gamma &\stackrel{(id,t)}{\to} & \Gamma \times Y
  }
  \,,
\end{displaymath}
with the horizontal morphism coming from a [[term]] $t : \Gamma \to Y$ in [[context]] $\Gamma$ and the vertical morphisms being the evident [[projection]], then the condition says that we may in a [[proposition]] $\phi$ substitute $t$ for $y$ before or after quantifying over $x$:

\begin{displaymath}
\sum_{x : X} \phi(x,t) \simeq (\sum_{x : X} \phi(x,y))[t/y]
 \,.
\end{displaymath}
\end{example}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{various}{}\subsubsection*{{Various}}\label{various}

\begin{itemize}%
\item The [[codomain fibration]] of any [[category]] with [[pullbacks]] is a bifibration, and satisfies the Beck--Chevalley condition at every pullback square.


\item If $C$ is a [[regular category]] (such as a [[topos]]), the bifibration $Sub(C) \to C$ of [[subobjects]] satisfies the Beck--Chevalley condition at every pullback square.


\item The [[family fibration]] $Fam(C)\to Set$ of any category $C$ with small sums satisfies the Beck--Chevalley condition at every pullback square in $Set$.


\item \href{double+coset#MackeyFormula}{Mackey's restriction formula} for group representations.



\end{itemize}
\hypertarget{PullbacksOfOpfibrations}{}\subsubsection*{{For categories of presheaves}}\label{PullbacksOfOpfibrations}

\begin{prop}
\label{BCForPresheavesOnPullbacksOfOpfibrations}\hypertarget{BCForPresheavesOnPullbacksOfOpfibrations}{}
If $\phi : D \to C$ is an [[opfibration]] of [[small categories]] and

\begin{displaymath}
\itexarray{
    D' &\stackrel{\beta}{\to}& D
    \\
    \downarrow^{\mathrlap{\psi}} && \downarrow^{\mathrlap{\phi}}
    \\
    C' &\stackrel{\alpha}{\to}& C
  }
\end{displaymath}
is a [[pullback]] diagram (in the \emph{1-category} [[Cat]]), and for $\mathcal{C}$ a [[category]] with all [[small colimits]], then the induced diagram of [[presheaf categories]]

\begin{displaymath}
\itexarray{
    [D', \mathcal{C}] &\stackrel{\beta^*}{\longleftarrow}& [D, \mathcal{C}]
    \\
    \uparrow^{\mathrlap{\psi}^*} 
      && 
    \uparrow^{\mathrlap{\phi}^*}
    \\
    [C', \mathcal{C}] &\stackrel{\alpha^*}{\longleftarrow}& [C,\mathcal{C}]
  }
  \,,
\end{displaymath}
satisfies the Beck-Chevalley condition: for $\psi_!$ and $\phi_!$ denoting the left [[Kan extension]] along $\psi$ and $\phi$, respectively, then we have a [[natural isomorphism]]

\begin{displaymath}
\psi_! \beta^* \simeq \alpha^* \phi_!
  \,.
\end{displaymath}
\end{prop}
(This is maybe sometimes called the \emph{projection formula}. But see at \emph{[[projection formula]]}.)

For this statement in the more general context of [[quasicategories]] see (\hyperlink{Joyal}{Joyal, prop. 11.6}).

\begin{proof}
Since $\phi$ is [[opfibration|opfibered]], for every object $c \in C$ the embedding of the [[fiber]] $\phi^{-1}(c)$ into the [[comma category]] $\phi/c$ is a [[final functor]]. Therefore the pointwise formula for the left [[Kan extension]] $\phi_!$ is equivalently given by taking the colimit over the fiber, instead of the comma category

\begin{displaymath}
\phi_1(X)_c \simeq \lim_{\underset{\phi^{-1}(c)}{\to}} X
  \,.
\end{displaymath}
Therefore we have a sequence of [[isomorphisms]]

\begin{displaymath}
\begin{aligned}
    (\psi_! \beta^* X)(c')
    & \simeq
    \lim_{\underset{\psi^{-1}(c')}{\to}} (X\circ \beta)
    \\
    & \simeq 
    \lim_{\underset{\phi^{-1}(\alpha(c'))}{\to}} X
    \\
    & \simeq
    (\alpha^* \phi_! X)(c')
  \end{aligned}
\end{displaymath}
all of them [[natural isomorphism|natural]] in $c'$.

\end{proof}
For an example that prop. \ref{BCForPresheavesOnPullbacksOfOpfibrations} may fail without the condition that $D \to C$ is an opfibration:

Consider $C=$ the [[interval category]] $(0\to 1)$, $D=C'=$ the [[terminal category]], $\phi=0$, $\alpha=1$, so that $D'=\emptyset$, but $\alpha^*\phi_!$ is the identity functor.

\hypertarget{proper_base_change_in_tale_cohomology}{}\subsubsection*{{Proper base change in \'e{}tale cohomology}}\label{proper_base_change_in_tale_cohomology}

For [[coefficients]] of [[torsion group]], [[étale cohomology]] satisfies Beck-Chevalley along [[proper morphisms]]. This is the statement of the \emph{[[proper base change theorem]]}. See there for more details.

\hypertarget{grothendieck_six_operations}{}\subsubsection*{{Grothendieck six operations}}\label{grothendieck_six_operations}

A kind of Beck-Chevalley condition appears in the axioms of Grothendieck's [[six operations]]. See there for more.

\hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages}

\begin{itemize}%
\item [[Benabou-Roubaud theorem]]

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

The original article is

\begin{itemize}%
\item [[Jean Bénabou]], [[Jacques Roubaud]], \emph{Monades et descente}, C. R. Acad. Sc. Paris, t. 270 (12 Janvier 1970), Serie A, 96--98, (\href{http://gallica.bnf.fr/ark:/12148/bpt6k480298g/f100}{link}, Biblioth\`e{}que nationale de France)

\end{itemize}
Discussion for [[subobject lattices]] is in

\begin{itemize}%
\item [[Saunders MacLane]], [[Ieke Moerdijk]], chapter IV.9 (around page 205) of \emph{[[Sheaves in Geometry and Logic]]}

\end{itemize}
Discussion for presheaf categories in the context of [[quasicategories]] ([[(infinity,1)-categories of (infinity,1)-presheaves]]) is in

\begin{itemize}%
\item [[André Joyal]], \emph{The Theory of Quasi-Categories and its Applications} (\href{http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf}{pdf})

\end{itemize}
[[!redirects Beck-Chevalley conditions]]

[[!redirects Beck-Chevalley\_Condition]] [[!redirects Beck-Chevalley Condition]] [[!redirects Beck–Chevalley condition]] [[!redirects Beck--Chevalley condition]] [[!redirects Beck condition]] [[!redirects Chevalley condition]] [[!redirects Beck-Chevalley property]] [[!redirects Beck–Chevalley property]] [[!redirects Beck--Chevalley property]]

[[!redirects Beck-Chevalley transformation]] [[!redirects Beck-Chevalley transformations]]



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