<sub>The following is a reasonably literal translation (made by me from the French original) of the published article: Jean Bénabou, Jacques Roubaud. _Monades et descente._ C. R. Acad. Sc. Paris, t. 270 (12 Janvier 1970), Série A, 96–98. [**Edit by DR Apr 2020**: additional notes moved to the bottom of the answer]</sub>
**Beginning of reasonably literal translation.**
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ALGEBRA. -- *Monads and descent*.
Note (${}^\ast$) by Jean Bénabou and Jacques Roubaud.
Transmitted by Henri Cartan.
<sub> By means of category theory, we interpret the 'descent data' in a simple and natural manner as 'algebras over a monad'.
This allows one, in very general situations, to recognize whether a morphism is a descent morphism or an effective descent morphism. </sub>
**1. The bifibrations of Chevalley, and descent.** -- In the following, $P\colon \textbf{M}\rightarrow \textbf{A}$ denotes a bifibrant functor $({}^1)$.
If $A$ is an object of $\textbf{A}$ the fiber over $A$ is denoted $\textbf{M}(A)$.
We assume that $A$ has all binary pullbacks.
1.1. *The monad associated with a morphism.* -- Let $a\colon A_1\rightarrow A_0$ be a morphism of $\textbf{A}$.
We denote the inverse image functor (resp. direct image functor) by
$a^\ast\colon\textbf{M}(A_0)\rightarrow\textbf{M}(A_1)\qquad$ (resp. $a_\ast\colon\textbf{M}(A_1)\rightarrow\textbf{M}(A_0)$)
and we denote the canonical natural transformations which make $a_\ast$ a left-adjoint of $a^\ast$ by
$\eta^a\colon\quad 1_{\textbf{M}(A_1)}\rightarrow a^\ast a_\ast;\qquad$ $\varepsilon^a\colon\quad a_\ast a^\ast\rightarrow 1_{\textbf{M}(A_0)}$
This adjunction defines $({}^2)$ a monad $\textbf{T}^a=(T^a,\mu^a,\eta^a)$ on $\textbf{M}(A_1)$, where
$T^a = a^\ast a_\ast\colon\quad \textbf{M}(A_1)\rightarrow\textbf{M}(A_1)$ ${}\qquad$ and ${}\qquad$ $\mu^a=a^\ast\varepsilon^a a_\ast\colon\quad T^a\circ T^a\rightarrow T^a$.
We denote by $\textbf{M}^a$ the category $\textbf{M}(A_1)^{(\textbf{T}^a)}$ of algebras of the monad $\textbf{T}^a$, and by
$U^{\textbf{T}a}\colon\quad\textbf{M}^a\rightarrow\textbf{M}(A_1)$ ${}\quad$ and ${}\quad$ $\Phi^a\colon\quad\textbf{M}(A_0)\rightarrow\textbf{M}^a$
the canonical functors.
1.2. *Chevalley condition.* $({}^3)$. -- We say $P$ is a *Chevalley functor* if the following property (C) is satisfied:
(C) For every commutative diagram
$\require{AMScd}$
\begin{CD}
\mathbf{M}'_0 @<\chi'<< \mathbf{M}'_1\\
@V k_0 V V @VV k_1 V\\
\mathbf{M}_0 @<<\chi< \mathbf{M}_1
\end{CD}
in $\textbf{M}$ whose image under $P$ is a pullback square of $\textbf{A}$, the following implication holds:
if $\chi$ and $\chi'$ are cartesian and $k_0$ is cocartesian, then $k_1$ is cocartesian.
1.3. *Characterization of descent data.* -- In the sequel, $P\colon\textbf{M}\rightarrow\textbf{A}$ is a Chevalley functor.
Let $a\colon A_1\rightarrow A_0$ be a morphism of $\textbf{A}$; let us write $A_2$ for the fibered product $A_1\times_{A_0}A_1$ and $a_1$ and $a_2$ for the canonical 'projections' of $A_2$ to $A_1$.
The property (C) permits to define, for each object $M_1$ de $\textbf{M}(A_1)$ a canonical bijection, 'natural' in $M_1$, of $\mathrm{Hom}_{\textbf{M}(A_2)}(a_1^\ast(M_1),a_2^\ast(M_1))$ onto $\mathrm{Hom}_{\textbf{M}(A_1)}(\textbf{T}^a(M_1),M_1)$, which we denote by $\varphi\mapsto K^a(\varphi)$.
Lemma. -- *A morphism $\varphi\colon a_1^\ast(M_1)\rightarrow a_2^\ast(M_1)$ such that $\textbf{P}(\varphi)=1_{A_2}$
is a descent datum if and only if $K^a(\varphi)$ is an algebra for the monad $\textbf{T}^a$.*
Let us write $\textbf{D}(a)$ for the category of descent data relative to $a$, and
$\Psi^a\colon\textbf{M}(A_0)\rightarrow\textbf{D}(a)$ ${}\qquad$ and ${}\qquad$ $U^a\colon\textbf{D}(a)\rightarrow\textbf{M}(A_1)$
for the canonical functors.
Theorem. -- *The correspondence $\varphi\mapsto K^a(\varphi)$ induces an equivalence of categories $K^a\colon\textbf{D}(a)\rightarrow \textbf{M}^a$ such that the following diagram commutes:*
[![enter image description here][3]][3]
Proposition 1. -- *The correspondence $\varphi\mapsto K^a(\varphi)$ is universal.*
More precisely, let $b_0\colon A_0'\rightarrow A_0$ be a morphism of $\textbf{A}$.
Base change gives rise to the diagram
[![enter image description here][4]][4]
in $\textbf{A}$.
If $M_1$ is an object of $\textbf{M}(A_1)$ and $\varphi\colon a_1^\ast(M_1)\rightarrow a_2^\ast(M_1)$ is a morphism of $\textbf{M}(A_2)$, then
$K^{a'}(b_2^\ast(\varphi)) = b_1^\ast(K^a(\varphi))$.
If one in particular takes $A_0'=A_1$ and $b_0=a$, then the following implication holds: if $\varphi$ is a descent datum, then $b_2^\ast(\varphi)$ is an effective descent datum.
The converse is true, since from the theorem and the proposition it follows that:
Corollary. -- *A morphism $\varphi\colon a_1^\ast(M_1)\rightarrow a_2^\ast(M_1)\in\textbf{M}(A_2)$ is a descent datum if and only if its inverse image $b_2^\ast(\varphi)$ via the canonical base change $b_0=a\colon A_0'=A_1\rightarrow A$ is an effective descent datum.*
This corollary allows to in the sequel avoid the use of the 'cocycle condition'
**2. First applications.** -- In view of the preceding theorem, a criterion of Beck (${}^2$) allows to give necessary and sufficient conditions for the functor $\Psi^a$ to be [severally] faithful, fully faithful, or an equivalence of categories, in terms of conditions which say that the inverse image functor $a^\ast$ reflects, or commutes with, certain cokernels.
Let us give some applications.
Proposition 2. -- *If the category $\textbf{M}(A_0)$ has all cokernel pairs, then the functor $\Psi^a$ has a left adjoint.*
Proposition 3. -- *The functor $\Psi^a$ is faithful if and only if $a^\ast$ is.*
Proposition 4. -- *If $a^\ast$ reflects cokernels, then $\Psi^a$ is fully faithful.*
In particular, if all the fibers of $\textbf{M}$ are abelian, then
$\text{
$\Psi^a$ faithful $\Leftrightarrow$ $\Psi^a$ fully faithful $\Leftrightarrow$ $a^\ast$ faithful
}$
Definition. -- We say that a morphism $a\colon A_1\rightarrow A_0$ is *faithfully flat* if the functor $a^\ast$ commutes with cokernels and reflects isomorphisms.
Proposition 5. -- *If $a\colon A_1\rightarrow A_0$ is faithfully flat, and if $\textbf{M}(A_0)$ has all cokernels, then $\Psi^a$ is an equivalence of categories.*
**3. First examples of Chevalley functors.**
3.1. If one takes for $\textbf{A}$ the dual of the category of commutative rings, and for $\textbf{M}$ the dual of the category of modules over [arbitrary] commutative rings, the evident functor $P\colon\textbf{M}\rightarrow\textbf{A}$ is a Chevalley functor.
3.2 If $\textbf{A}$ is a category with all binary pullbacks, and if $\textbf{M} = \mathrm{Fl}(\textbf{A})$ denotes the arrow category of $\textbf{A}$, the 'codomain' functor $\textbf{M}\rightarrow\textbf{A}$ is a Chevalley functor.
3.3. If $\textbf{P}\colon\textbf{M}\rightarrow\textbf{A}$ and $\textbf{Q}\colon\textbf{N}\rightarrow\textbf{M}$ are Chevalley functors, their composite $P\circ Q$ is.
3.4. If $\textbf{P}\colon\textbf{M}\rightarrow\textbf{A}$ is a Chevalley functor, and if $\textbf{II}$ [I think this is simply a misprint in the C.R.Acad.Sc. paper, and should rather be $\textbf{I}$] is any category, then the functor $P^{\textbf{I}}\colon \textbf{M}^{\textbf{I}}\rightarrow\textbf{A}^{\textbf{I}}$ is a Chevalley functor.
3.5. If in a pullback square
\begin{CD}
\mathbf{M} @<\chi'<< f^*(\mathbf{M})\\
@V P V V @VV f^*(P) V\\
\mathbf{A} @<<f< \mathbf{X}
\end{CD}
the category $\textbf{X}$ has all finite pullbacks, then the functor $f$ commutes with pullbacks, and the functor $P$ is a Chevalley functor, hence $f^\ast(P)$ is a Chevalley functor.
In a later publication, we will give further examples of Chevalley categories,
and also more precise criteria which allow to recognize whether $\Psi^a$ is faithful, fully faithful,
or an equivalence of categories in situations where the fibers of $\textbf{M}$ are algebraic categories
(categories of modules, for example).
(${}^\ast$) Session of January 5, 1970.
(${}^1$) A. Grothendieck, *Catégories fibrées et descente.* (Séminaire Bourbaki, 1959).
(${}^2$) Linton, *Applied Functorial semantics.* II, Springer lecture Notes no 80, 1969.
(${}^3$) Chevalley, *Séminaire sur la descente 1964--1965* (unpublished).
[postal adresses at end of the article omitted]
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**End of reasonably literal translation.**
Addition 11 November 2017. The following is the review of op. cit. in Zbl 0287.18007:
[![enter image description here][6]][6]
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<sub> While I do think that I got it mostly right, and chose the modern technical terms conscientiously, I do not take any responsibility for this translation, neither for its form nor its content, and there might be a few places where it might be possible to improve the translation.
Do not take it to be a 'sworn translation', or anything like that.
If in doubt, consult the original article.</sub>
<sub>(0) I do not have anything against someone developing this answer further, *if* it is an on-topic improvement.</sub>
<sub>(1) I am mentioning (0) since:</sub>
<sub>(1.0) Translating is an [art and a craft][1] of its own, with its own professional associations customs, and translating well takes considerable work and knowledge of both the source- and target-language and the subject matter, and many translators would not like to see their work changed by others, so (0) does not go without saying. </sub>
<sub>(1.1) It will not be possible for me to henceforth be responsible for this translation, *let alone even check this thread at regular intervals*.</sub>
<sub>Please consider my translation to be in the public domain, and please do not 'ping' me too much about it. </sub>
<sub> Please go ahead *if* you think you are sure that a relevant improvement should be made.</sub>
<sub>(2) In contrast to (0), I am not sure whether the *authors* of the translated article have something against expanding the translation here.
The reason for that is that I think that then the result of the process will not be literal translation anymore.
The boundaries between 'literal translation of' and 'some-sort-of-group-blog-exposition of' a given mathematical classic will blur.
Maybe one should keep these two genres separate, I am not sure.</sub>
<sub>(3) Why I did I write this: simply because this question was asked and I think this is an answer to the OP. Moreover, I myself have to work with similar things recently, and I had planned to understand descent-theory a little better anyway.</sub>
<sub>And now there appeared this request which harmonized with that. Incidentally, the comment to the effect that the author of the blog-post has been emailed the OP, which may lead to the OP getting another translation soon, was read by me only after my translation was almost finished.</sub>
<sub>Moreover, I can find little wrong with there being several translations of classics;</sub> <sub>there are several translations of, say, the classics of Latin poetry, too, so why should there not be multiple translations of French classics of category theory?</sub>
[1]: https://en.wikipedia.org/wiki/American_Translators_Association
[2]: https://i.sstatic.net/wsdie.png
[3]: https://i.sstatic.net/nkags.png
[4]: https://i.sstatic.net/aKYBo.png
[5]: https://i.sstatic.net/xVWUf.png
[6]: https://i.sstatic.net/WxCtr.png