This is about a famous theorem from
Jean Bénabou, Jacques Roubaud, Monades et descente, C. R. Acad. Sc. Paris, t. 270 (12 Janvier 1970), Serie A, 96–98, (link, Bibliothèque nationale de France)
Zoran: a file with my few years old English translation will be posted and linked in few days or weeks
relating descent via fibered categories (as in SGA I or FGA explained) to monadic descent. There has been some historical discussion on this in the category list; Zoran’s response is here.
A functor $P : F\to A$ is a Grothendieck opfibration if $P^{op}:F^{op}\to A^{op}$ is a Grothendieck fibration, and a functor $P:F\to A$ is a bifibration if $P$ is both a Grothendieck opfibration and fibration (no additional compatibility asked!). Thus we can talk about cartesian and cocartesian arrows in $F$.
In a bifibered category, automatically for any morphism $a:A_1\to A_0$ in the base, the “inverse image” (or “pullback” or “restriction”) functor $a^*:F(A_0)\to F(A_1)$ is right adjoint to the “pushforward” functor $a_!:F(A_1)\to F(A_0)$; with unit $\eta^a : Id_{F(A_1)} \to a^* a_!$ and counit $\epsilon^a : a_! a^* \to Id_{F(A_0)}$.
(Note that in topos theory and algebraic geometry, functors $a^*$ called “inverse images” usually have right adjoints $a_*$. This situation can be reconciled with the setup of bifibrations either by taking fiberwise opposites, so that left and right adjoints are switched, or by taking opposites of both the base and total categories, so that the direct and inverse images are switched. However, there are also many bifibrations arising in other contexts in which $a^*$ has both a left adjoint $a_!$ and a right adjoint $a_*$, although the latter cannot then be described cleanly in fibrational terms.)
The adjunction $a_!\dashv a^*$ generates a monad $\mathbf{T}^a=(T^a,\mu^a,\eta^a)$ in the usual way: the functor is $T^a = a^* a_!\colon F(A_1)\to F(A_1)$, the multiplication is $\mu^a = a^* \epsilon^a a_!\colon T^a \circ T^a \to T^a$, and the unit is just the unit of the adjunction. Denote by $F^a$ the Eilenberg–Moore category $F(A_1)^{\mathbf{T}^a}$ of modules (algebras) over the monad $\mathbf{T}^a$, with canonical forgetful functor $U^{\mathbf{T}} \colon F^a \to F(A_1)$ and canonical comparison functor $\Phi^a : F(A_0) \to F^a$.
Now we assume that $A$ has pullbacks, and that $P$ satisfies what is nowadays called the Beck-Chevalley property, namely that for each commutative square
in $F$ such that its image in $A$ is a pullback square, if $\chi$ and $\chi'$ are cartesian and $k_0$ is cocartesian then $k_1$ is cocartesian.
Mike Shulman: Is that really correct? I would have thought it would be “if $\chi$ and $\chi'$ are cartesian and $k_1$ is opcartesian, then $k_0$ is also opcartesian.
Zoran: I have to think, I got rusty in these issues.
Beren Sanders: I believe Mike is correct; I’ve corrected the statement accordingly.
An equivalent way to state the condition is that for any pullback square
in $A$, the canonical transformation $c_! a^* \to b^* d_!$ is an isomorphism. In the Bénabou–Roubaud paper this is called the Chevalley property and said to make $P$ into a Chevalley functor.
Denote by $A_2 :=A_1\times_{A_0}A_1$ the pullback of $a$ along itself, with the canonical projections $a_1,a_2\colon A_2\to A_1$. Now consider the lift of the cartesian square defining $A_2$ to $F$ in such a way that $a_1$ is lifted to a cartesian arrow, $a_2$ to a cocartesian arrow, and $a$ to a cocartesian arrow. Then by the universality there is a lift of $a$ completing the square, and by the Beck–Chevalley property it is cartesian. Together with the isomorphism given by adjunction this gives a morphism
One checks that an invertible morphism $\phi\colon a_1^*(M_1)\to a_2^*(M_1)$ satisfies the cocycle equation (making it into a descent datum) iff $K^a(\phi)$ is an action of $\mathbf{T}^a$ on $M_1$, and similarly for the unitality axiom.
Denote by $Desc(a)$ the category of descent data for the fibration $P$ along the morphism $a$; it comes with canonical functors
and
The Bénabou–Roubaud theorem asserts that this induces an equivalence of categories between $Desc(a)$ and $F^a$.
In addition, this equivalence satisfies some naturality properties, including that it commutes appropriately with the canonical functors to the fibers $F(A_0)$ and $F(A_1)$. Combining this theorem with Beck’s monadicity theorem, it becomes a practical tool for establishing a descent property in bifibrations, with variants in some other setups (to be covered later).
There are several characterizations of a Beck-Chevalley property for bifibrations:
(Duško Pavlović, in Category theory Como 1990, LNM 1488, Springer 1991)
Let $p: F\to B$ be a bifibration, $Q = (f,g,s,t)$ a square in $B$ such that $f\circ g = s\circ t$, and $\Theta = (\phi, \gamma, \sigma, \theta)$ a square in $F$ such that $\phi\circ\gamma=\sigma\circ\theta$, with $p(\phi)=f$, $p(\gamma)=g$, $p(\sigma)=s$ and $p(\theta)=t$. The following conditions are equivalent:
a) if $\theta$ and $\phi$ are cartesian and if $\sigma$ is cocartesian then $\gamma$ must be cocartesian;
b) if $\sigma$ and $\gamma$ are cocartesian and if $\theta$ is cartesian then $\phi$ must be cartesian;
c) if $\theta$ is cartesian and if $\sigma$ is cocartesian then $\phi$ is cartesian iff $\gamma$ is cocartesian.
If some inverse image functors $f^*$ and $t^*$ and some direct image functors $g_!$ and $s_!$ are chosen, then every square $\Theta$ over $Q$ satisfies conditions (a-c) iff there is a canonical natural isomorphism
d) $f^*\circ s_! \cong g_! \circ t^*$.