# What is descent theory?

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,

• What is descent theory in general? And what are it's unifying abstract patterns?

• 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).

• Could you make some examples of theorems which are "typical" of descent theory? And also mention the most important and well known theorems?