nLab Bénabou-Roubaud theorem




The theorem (Bénabou-Roubaud 70) identifies the Eilenberg-Moore category of algebras over the monad induced from a base change adjoint triple for some bifibration satisfying the Beck-Chevalley condition with the category of descent data along this morphism. This is the basis for the monadic reformulation of descent theory: monadic descent.


A functor P:FAP : F\to A is a Grothendieck opfibration if P op:F opA opP^{op}:F^{op}\to A^{op} is a Grothendieck fibration, and a functor P:FAP:F\to A is a bifibration if PP is both a Grothendieck opfibration and fibration (no additional compatibility asked!). Thus we can talk about cartesian and cocartesian arrows in FF.

Given a bifibration P:FAP : F\to A, automatically for any morphism a:A 1A 0a:A_1\to A_0 in AA the “inverse image” (or “pullback” or “restriction”) functor a *:F(A 0)F(A 1)a^*:F(A_0)\to F(A_1) is right adjoint to the “pushforward” functor a !:F(A 1)F(A 0)a_!:F(A_1)\to F(A_0); with unit η a:Id F(A 1)a *a !\eta^a : Id_{F(A_1)} \to a^* a_! and counit ϵ a:a !a *Id F(A 0)\epsilon^a : a_! a^* \to Id_{F(A_0)}.

(Note that in topos theory and algebraic geometry, functors a *a^* called “inverse images” usually have right adjoints a *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 *a^* has both a left adjoint a !a_! and a right adjoint a *a_*, although the latter cannot then be described cleanly in fibrational terms.)

The adjunction a !a *a_!\dashv a^* generates a monad T a=(T a,μ a,η a)\mathbf{T}^a=(T^a,\mu^a,\eta^a) in the usual way: the functor is T a=a *a !:F(A 1)F(A 1)T^a = a^* a_!\colon F(A_1)\to F(A_1), the multiplication is μ a=a *ϵ aa !:T aT aT a\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 aF^a the Eilenberg–Moore category F(A 1) T aF(A_1)^{\mathbf{T}^a} of modules (algebras) over the monad T a\mathbf{T}^a, with canonical forgetful functor U T:F aF(A 1)U^{\mathbf{T}} \colon F^a \to F(A_1) and canonical comparison functor Φ a:F(A 0)F a\Phi^a \colon F(A_0) \to F^a.

Now we assume that AA has pullbacks, and that PP satisfies what is nowadays called the Beck-Chevalley property, namely that for each commutative square

N 0 χ N 1 k 0 k 1 M 0 χ M 1 \array{ N_0 & \stackrel{\chi'}\leftarrow & N_1 \\ {}^{\mathllap{k_0}}\downarrow && \downarrow^{\mathrlap{k_1}} \\ M_0 &\stackrel{\chi}\leftarrow & M_1 }

in FF such that its image in AA is a pullback square, if χ\chi and χ\chi' are cartesian and k 0k_0 is cocartesian then k 1k_1 is cocartesian.

An equivalent way to state the condition is that for any pullback square

x a y c d z b w \array{ x & \overset{a}{\to} & y \\ {}^{\mathllap{c}} \downarrow && \downarrow^{\mathrlap{d}} \\ z & \underset{b}{\to} & w }

in AA, the canonical transformation c !a *b *d !c_! a^* \to b^* d_! is an isomorphism. In the Bénabou–Roubaud paper this is called the Chevalley property and said to make PP into a Chevalley functor.

Denote by A 2:=A 1× A 0A 1A_2 :=A_1\times_{A_0}A_1 the pullback of aa along itself, with the canonical projections a 1,a 2:A 2A 1a_1,a_2\colon A_2\to A_1. Now consider the lift of the cartesian square defining A 2A_2 to FF in such a way that a 1a_1 is lifted to a cartesian arrow, a 2a_2 to a cocartesian arrow, and aa to a cocartesian arrow. Then by the universality there is a lift of aa completing the square, and by the Beck–Chevalley property it is cartesian. Together with the isomorphism given by adjunction this gives a morphism

K a:Hom F(A 2)(a 1 *(M 1),a 2 *(M 1))Hom F(A 1)(T a(M 1),M 1). K^a: Hom_{F(A_2)}(a_1^*(M_1),a_2^*(M_1))\to Hom_{F(A_1)}(T^a(M_1),M_1).

One checks that an invertible morphism ϕ:a 1 *(M 1)a 2 *(M 1)\phi\colon a_1^*(M_1)\to a_2^*(M_1) satisfies the cocycle equation (making it into a descent datum) iff K a(ϕ)K^a(\phi) is an action of T a\mathbf{T}^a on M 1M_1, and similarly for the unitality axiom.

The theorem

Denote by Desc(a)Desc(a) the category of descent data for the fibration PP along the morphism aa; it comes with canonical functors

Ψ a:F(A 0)Desc(a) \Psi^a\colon F(A_0)\to Desc(a)


U a:Desc(a)F(A 1). U^a\colon Desc(a)\to F(A_1) \,.

The Bénabou–Roubaud theorem asserts that this induces an equivalence of categories between Desc(a)Desc(a) and F aF^a.

In addition, this equivalence satisfies some naturality properties, including that it commutes appropriately with the canonical functors to the fibers F(A 0)F(A_0) and F(A 1)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:FBp: F\to B be a bifibration, Q=(f,g,s,t)Q = (f,g,s,t) a square in BB such that fg=stf\circ g = s\circ t, and Θ=(ϕ,γ,σ,θ)\Theta = (\phi, \gamma, \sigma, \theta) a square in FF such that ϕγ=σθ\phi\circ\gamma=\sigma\circ\theta, with p(ϕ)=fp(\phi)=f, p(γ)=gp(\gamma)=g, p(σ)=sp(\sigma)=s and p(θ)=tp(\theta)=t. The following conditions are equivalent:

a) if θ\theta and ϕ\phi are cartesian and if σ\sigma is cocartesian then γ\gamma must be cocartesian;

b) if σ\sigma and γ\gamma are cocartesian and if θ\theta is cartesian then ϕ\phi must be cartesian;

c) if θ\theta is cartesian and if σ\sigma is cocartesian then ϕ\phi is cartesian iff γ\gamma is cocartesian.

If some inverse image functors f *f^* and t *t^* and some direct image functors g !g_! and s !s_! are chosen, then every square Θ\Theta over QQ satisfies conditions (a-c) iff there is a canonical natural isomorphism

d) f *s !g !t *f^*\circ s_! \cong g_! \circ t^*.


The original article:

See also:

  • George Janelidze, Walter Tholen, Facets of Descent I, Applied Categorical Structures 1994, Volume 2, Issue 3, pp 245-281

  • George Janelidze, Walter Tholen, Facets of Descent II, Applied Categorical Structures September 1997, Volume 5, Issue 3, pp 229-248

  • X. Guo, thesis, p. 58 of Monadicity, purity and descent equivalence, pdf

There has been some historical discussion on this in the category list; Zoran’s response is here.

The following reference presents a proof for a (generalization) of the Bénabou-Roubaud Theorem (pages 435 and 436):

Last revised on November 14, 2023 at 15:48:40. See the history of this page for a list of all contributions to it.