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.
Suppose given a commutative square (up to isomorphism) of functors:
in which $f^*$ and $h^*$ have left adjoints $f_!$ and $h_!$, respectively. (The classical example is a Wirthmüller context.) Then the natural isomorphism that makes the square commute
has a mate
defined as the composite
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,
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.
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.
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 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.
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.
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 (-1)-truncation, possibly) as existential quantification.
In terms of this the Beck-Chevalley condition says that if the above diagram is a pullback, then substitution commutes with existential quantification
Consider the diagram of contexts
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 $x$:
If $\phi : D \to C$ is an opfibration of small categories and
is a pullback diagram (in the 1-category Cat), then the induced diagram of presheaf categories
for $\mathcal{C}$ a category with all small colimits, satisfies the Beck-Chevalley condition: for $\psi_!$ and $\phi_!$ denoting the left Kan extension along $\psi$ and $\phi$, respectively, we have a natural isomorphism
This is sometimes called the projection formula.
Since $\phi$ is 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
Therefore we have a sequence of isomorphisms
all of them natural in $c'$.
Originally, the Beck-Chevalley condition was introduced in (Bénabou-Roubaud, 1970) 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 $\mathbf{X} \to \mathbf{B}$ where $\mathbf{B}$ has pullbacks satisfies the Chevalley condition iff for every commuting square
in $\mathbf{X}$ over a pullback square in the base $\mathbf{B}$ where $\varphi$ is 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.
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 $Sub(C) \to C$ of subobjects satisfies the Beck–Chevalley condition at every pullback square.
The family fibration? $Fam(C)\to Set$ of any category $C$ with small sums satisfies the Beck–Chevalley condition at every pullback square in $Set$.
For coefficients of torsion group, étale cohomology satisfies Beck-Chevalley along proper morphisms. This is the statement of the proper base change theorem. See there for more details.
A kind of Beck-Chevalley condition appears in the axioms of Grothendieck’s six operations. See there for more.
A textbook reference is for instance section IV.9 (page 205) of