nLab
Beck-Chevalley condition

Idea

The Beck–Chevalley condition, also sometimes called just the Beck condition or the Chevalley condition, is a “commutation of adjoints” property that holds in many “change of base” situations.

Definition

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

$$\begin{array}{ccc}& \stackrel{f^{*}}{\to }& \\ {}^{g^{*}}\downarrow & & {\downarrow }^{k^{*}}\\ & \underset{h^{*}}{\to }& \end{array}$$\array{ & \overset{f^*}{\to} & \\ ^{g^*}\downarrow && \downarrow^{k^*}\\ & \underset{h^*}{\to} & }

in which $f^{*}$ and $h^{*}$ have left adjoints $f_{!}$ and $h_{!}$, respectively. Then the natural isomorphism that makes the square commute has a mate

$$h_{!}{k}^{*}\to g^{*}{f}_{!}$$h_! k^* \to g^* f_!

defined as the composite

$$h_{!}{k}^{*}\stackrel{\eta }{\to }{h}_{!}{k}^{*}{f}^{*}{f}_{!}\stackrel{\cong }{\to }{h}_{!}{h}^{*}{g}^{*}{f}_{!}\stackrel{ϵ}{\to }{g}^{*}{f}_{!}.$$h_! k^* \overset{\eta}{\to} h_! k^* f^* f_! \overset{\cong}{\to} h_! h^* g^* f_! \overset{\epsilon}{\to} g^* f_! .

We say the original square satisfies the 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 Beck–Chevalley condition in this case,

Left and right Beck–Chevalley 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.

Dual Beck–Chevalley condition

If $f^{*}$ and $h^{*}$ have 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.

“Local” Beck–Chevalley condition

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 ${\eta }_x:x\to f^{*}{f}_{!}x$ which is initial in the comma category $(x/f^{*})$. Then we have $k^{*}(\eta ):k^{*}x\to k^{*}{f}^{*}{f}_{!}x\to h^{*}{g}^{*}{f}_{!}x$, and we say that the square satisfies the 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 asumed 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.

Bifibrations

Originally, the condition was introduced by B'enabou and Roubaud in 1970 in their classical paper “Monades et Descente” for bifibrations over a base category with pullbacks. In loc.cit. they call this condition Chevalley condition because he introduced it in his 1964 seminar.

A bifibration $X\to B$ where $B$ has pullbacks satisfies the Chevalley condition iff for every commuting square

$$\begin{array}{ccc}& \stackrel{{\psi }^{\prime }}{\to }& \\ {\downarrow }^{{\phi }^{\prime }}& & {\downarrow }^{\phi }\\ & \underset{\psi }{\to }& \end{array}$$\array{ & \overset{\psi^\prime}{\rightarrow} & \\ \downarrow^{\varphi^\prime} && \downarrow^{\varphi}\\ & \underset{\psi}{\rightarrow} & }

in $X$ over a pullback square in the base $B$ where $\phi$ is cartesian and $\psi$ is cocartesian it holds that ${\phi }^{\prime }$ 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 of cleavages.

A fibration $P$ has products satisfying the Chevalley condition iff the opposite fibration $P^{\mathrm{op}}$ 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.

Examples

• The codomain fibration of any category with pullbacks is a bifibration, and satisfies the Beck–Chevalley condition at every pullback square.

• If $C$ is a regular category (such as a topos), the bifibration $\mathrm{Sub}(C)\to C$ of subobjects satisfies the Beck–Chevalley condition at every pullback square.

• The family fibration? $\mathrm{Fam}(C)\to \mathrm{Set}$ of any category $C$ with small sums satisfies the Beck–Chevalley condition at every pullback square in $\mathrm{Set}$.

References

For instance section IV.9 (page 205) of

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic