This question is probably too vague for experts, but I really don't know how to avoid it.


I've read in several places that under mild conditions, a morphism is an effective descent morphism iff the base-change functor it induces is monadic.


Now, I don't really know anything about descent and I'm trying to probe around and figure out how to start learning about it. I think of monads in terms of algebraic theories. Descent on the other is supposedly of geometric nature; a formalism which generalizes familiar gluing over open subsets to a more general setting without a spatial topology. I can't imagine how or why these two notions should be related.


What lies at the core of the relationship between monadicity - a seemingly algebraic and concrete notion - and (effective) descent (morphisms)?



Why are monadicity and descent related?
Probably the reason "monadicity" gets connected with descent (and the associated terminology of descent theory) is because of its relevance to the question of descent for rings. 


If you're talking about a morphism of rings $\phi:A\to B$ there is a functor $-\otimes\_AB:Mod\_A\to Mod\_B$. Then you can ask the question: Can I recover the category $Mod\_A$ from $Mod\_B$? This question is answered by knowing whether or not $-\otimes\_AB$ is comonadic. In particular, this tells us that $Mod\_A$ is equivalent to the category of $ \phi^\ast\circ(-\otimes\_AB)$-comodules in $Mod\_B$. It takes a little bit of work, but as an exercise for really understanding descent, it may be useful to prove that in this case, the category of comodules for this comonad is equivalent to the category of $B\otimes\_AB$-comodules, where $B\otimes\_AB$ is a $B$-coring with comultiplication $$B\otimes\_AB\cong B\otimes\_AA\otimes\_AB\to B\otimes\_AB\otimes\_AB\cong B\otimes\_AB\otimes\_B B\otimes\_AB.$$


This object is also sometimes called the descent coring, and for reasons that will hopefully become clear in the next paragraph, its comodules are often referred to as descent data.


With a bit of finagling one can prove that a $B\otimes\_AB$-comodule structure on a $B$-module $N$ (so in the case that $-\otimes\_AB$ is comonadic we know that $N\cong M\otimes\_AB$ for a unique $A$-module $M$) is the same thing as a "cocycle condition" satisfying isomorphism $$M\otimes\_AB \cong M\otimes\_B B\otimes\_AB\overset{\sim}\to B\otimes\_AB\otimes\_B M\cong B\otimes\_AM.$$ The basic idea is to apply the adjunction $$Hom\_B(M,B\otimes\_AM)\cong Hom\_{B\otimes\_AB}(M\otimes\_AB,B\otimes\_AM).$$


And so this is the thing that lets us take a geometric perspective. If we consider the equivalent problem of whether or not we can recover the category of quasicoherent sheaves on $Spec(A)$ from the category of quasicoherent sheaves on $Spec(B)$, with adjunction $\phi^\ast:QC\_{Spec(A)}\rightleftarrows QC\_{Spec(B)}:\phi\_\*$ then the descent data (as defined above for modules) for a sheaf $\mathcal{F}$ on $Spec(B)$ is an isomorphism between the two ways of pulling back $\mathcal{F}$ to $Spec(B)\times\_{Spec(A)}Spec(B).$ This, as you may recall, is precisely the way that we typically state "gluing" conditions. In other words, if a sheaf supports such an isomorphism, it lives in the coequalizer (in categories, so necessarily the coequalizer computed as a 2-colimit) of the diagram $$QC\_{Spec(B)}\leftleftarrows QC\_{Spec(B)\times\_{Spec(A)} Spec(B)}\Lleftarrow QC\_{Spec(B)\times\_{Spec(A)} Spec(B)\times\_{Spec(A)} Spec(B)}.$$


The only further thing maybe to mention is that usually geometric-type descent is stated in terms of covers $X\overset{f\_i}\leftarrow\{U\_i\}$ but we can do everything above by just considering $X\overset{\coprod f\_i}\leftarrow\coprod U\_i.$


So now whenever you've got any kind of monad or comonad, you can ask about "descent" for that monad, which is really just a question of whether or not you can recover some category from some other category of (co)monadic (co)modules.


