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.

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.

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.

Bifibrations

A common situation in which this occurs is when we have a bifibration, with the functors $f^{*}$, $g^{*}$, etc. being the “pullback” functors coming from the cartesian liftings of a commutative square

$$\begin{array}{ccc}& \stackrel{f}{\leftarrow }& \\ {}^g\uparrow & & {\uparrow }^k\\ & \underset{h}{\leftarrow }& \end{array}$$\array{ & \overset{f}{\leftarrow} & \\ ^g\uparrow && \uparrow^k\\ & \underset{h}{\leftarrow} & }

in the base category. In this case one also says that this commutative square “downstairs” has the Beck–Chevalley property. Frequently this property holds for all pullback squares in the base category. Note that since the transpose of a pullback square is a pullback square, in this case there is no left/right ambiguity.

According to the Benabou–Roubaud theorem, the Beck–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 $C$ satisfies the Beck–Chevalley condition at every pullback square in $\mathrm{Set}$.

References

• See Mac Lane and Moerdijk, section IV.9 (page 205).