I read the article in wikipedia, but I didn't find it totally illuminating. As far as I've understood, essentially you have a morphism (in some probably geometrical category) $Y \rightarrow X$, where you interpret $Y$ as being the "disjoint union" of some "covering" (possibly in the Grothendieck topology sense) of $X$, and you want some object $\mathcal{F'}$ defined on $Y$ to descend to an object $\mathcal{F}$ defined on $X$ that will be isomorphic to $\mathcal{F}'$ when pullbacked to $Y$ (i.e. "restricted" to the patches of the covering). To do this you have problems with $Y\times_{X}Y$, which is interpreted as the "disjoint union" of all the double intersections of elements of the cover.
I'm aware of the existence of books and notes on -say- Grothendieck topologies and related topics (that I will consult if I'll need a detailed exposition), but I would like to get some ideas in a nutshell, with some simple and maybe illuminating examples from different fields of mathematics.
I also know that there are other MO questions related to descent theory, but I think it's good that there's a (community wiki) place in which to gather instances, examples and general picture.
So,
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What is descent theory in general? And what are it's unifying abstract patterns?
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In which fields of mathematics does it appear or is relevant, and how does it look like in each of those fields? (I'm mostly interested in instances within algebraic geometry, but having some picture in other field would be nice).
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Could you make some examples of theorems which are "typical" of descent theory? And also mention the most important and well known theorems?
Best Answer
Suppose we are given some category (or higher category) of "spaces" in which each space $X$ is equipped with a fiber, i.e. a category $C_X$ of objects of some type over it. For example, a space can be a smooth manifold and the fiber is the category of vector bundles over it; or a space is an object of the category dual to the category of rings and the fiber is its category of left modules. Given a map $f: Y\to X$, one often has an induced functor $f^* : C_X\to C_Y$ (pullback, inverse image functor, extension of scalars). The basic questions of classical descent theory are:
When an object $G$ in $C_Y$ is in the image via $f^*$ of some object in $C_X$ ?
Classify all forms of object $G\in C_Y$, that is find all $E\in C_X$ for which $f^*(E)\cong G$.
Grothendieck introduced pseudofunctors and fibred categories to formalize an ingenious method to deal with descent questions. He introduces additional data on an object $G$ in $C_Y$ to have a chance of determining an isomorphism class of an object in $C_X$. Such an enriched object over $X$ is called a ``descent datum''. $f$ is an effective descent morphism if the morphism $f$ induces a canonical equivalence of the category of the descent data (for $f$ over $X$) with $C_X$. It is a nontrivial result that in the case of rings and modules, the effective descent morphisms are preciselly pure morphisms of rings. Grothendieck's flat descent theory tells a weaker result that faithfully flat morphisms are of effective descent. In algebraic situations one often introduces a (co)monad $T_f : C_X\to C_X$ (say with the multiplication $\mu: T_f \circ T_f \to T_f$) induced by the morphism $f$. The category of descent data is then nothing else than the Eilenberg-Moore category $T_f-\mathrm{Mod}$ of (co)modules (also called (co)algebras) over $T_f$. Then, by the definition, $f$ is of an effective descent if and only if the comparison map (defined in the (co)monad theory) between $C_X$ and $T_f-\mathrm{Mod}$ is an equivalence. Several variants of Barr-Beck theorem give conditions which are equivalent or (in some variants) sufficient to the comparison map for a monad induced by a pair of adjoint functors being an equivalence. Generically such theorems are called monadicity (or tripleability) theorems. One can describe most of (but not all) situations of 1-categorical descent theory via the monadic approach.
There are numerous generalizations of monadicity theorems, higher cocycles and descent, both in monadic and in fibered category setup in higher categorical context (Giraud, Breen, Street, K. Brown, Hermida, Marmolejo, Mauri-Tierney, Jardine, Joyal, Simpson, Rosenberg-Kontsevich, Lurie...); the theory of stacks, gerbes and of general cohomology is almost the same as the general descent theory, in a point of view.
For examples, it is better to consult the literature. It takes a while to treat them.