Monadic descent is a way to encode descent of fibered categories (equivalently, by the Grothendieck construction, of pseudofunctors) that have the special property that they are bifibrations. This allows to use algebraic tools, notably monads and related structures from universal algebra:
a bifibration comes naturally equipped not only with a notion of pullback, but also of pushforward. Combined these provide pull-push-monads that may be used to encode the descent property of the fibration.
A morphism in the base induces an adjunction where
and we ask whether is a monadic functor.
The adjunction, with unit and counit , induces the monad in via , with unit . The monad generates the Eilenberg–Moore category of modules (= algebras) over , that is the pairs where and satisfies the action axioms and .
We think of this data as follows:
is the category of some type of objects on some space;
is the category of this type of object on a cover of that space
is the category of descent data for such objects, with respect to that cover.
There is a comparison functor
We say that the functor is monadic or more informally that is monadic over if the comparison functor is an equivalence of categories. Sometimes one is interested in a weaker condition: whether the comparison functor is fully faithful.
The main theorem is Beck’s monadicity theorem.
Given a Grothendieck bifibration and a morphism in the base category , one can choose a direct image and an inverse image functor , which form an adjunction . Under some conditions (see the Bénabou–Roubaud theorem), the morphism is an effective descent morphism (with respect to as a fibered category) iff the comparison functor for the monad induced by the adjunction is monadic.
One of the most basic examples of bifibrations are codomain fibrations . Accordingly, monadic descent applied to codomain fibrations archetypically exhibits the nature of monadic descent. We therefore spell out this example is some detail.
An object in a codomain fibration over is a morphism , hence a bundle in , in the most general sense of bundle. Therefore monadic descent with respect to codomain fibrations encodes descent of bundles.
Other examples of monadic descent often find a useful interpretation when relating them back to monadic descent for codomain fibrations. For instance (co)monadic descent for Sweedler corings, discussed below, finds a natural geometric interpretion this way (as discussed in detail there).
equipped with a transition function between its two pullbacks to double
which satisfies on the usual cocycle condition.
Monadic methods can be applied to the study of descent of structures that cannot only be pulled back, such as principal bundles, but that can also be pushed forward, such as vector bundles (with suitable care taken) or more generally modules over functions rings (discussed at Sweedler coring).
Given a principal bundle (a topological one, say, i.e. a morphism in Top) and a morphism of base spaces , the would-be candidate for the push-forward of along is simply the composite map , regarded as a total space living over .
While that always exists as such, it will in general not be a principal bundle anymore: the fibers of over points consist of many copies of the fibers of over points in . Hence the shape of the fibers may change drastically as we push bundles forward.
So principal bundles do have a canonical notion of push-forward, but it leads outside the category of principal bundles and lands generally in some overcategory.
On the other hand, as we will see in detail below, if we take a principal bundle and
first push it forward in this generalized sense to an object in the overcategory
and then pull back the result of that again along the result, while still not a principal bundle, is the total space of the bundle pulled back to the first term in the Čech nerve of . This pullback is of central interest in the description of the geometric descent property of the bundle.
But the composite operation of pushforward of overcategories
followed by pullback
So by regarding principal bundles more generally as just objects in the overcategory we make the tools of monadic descent applicable to them.
The direct image operation associated to a morphism in is the functor
obtained by postcomposition with , which sends to the composite in , regarded as an object of .
The inverse image operation associated to is the functor
To that end, let
be an algebra over our monad. In components this is an object equipped with a morphism .
The object is given by
first pushing forward along to the object .
then pulling this back along to yield the left vertical morphism in
This pullback along a composite of morphisms may be computed as two consecutive pullbacks. The first one is
which is the first term in the Čech nerve of . So the total pullback is the pullback to :
Therefore the action of our monad on is given in by a morphism
As an example, think of this in the context with coming from an open cover of with , and with a trivial -principal bundle for some group . Then the space is the union of double intersection of covering patches, and is to be thought of as the trivial -principal bundle over , restricted to the intersection. In this case our morphism acts as
and thus maps the trivial -bundle over on the intersection with the trivial -bundle over . So it is a transition function. If this is a -equivariant, it may be part of the descent datum for the -principal bundle.
Now we consider monadic descent along morphisms that happen to be -principal bundles, for some group object . When considered with respect to the codomain fibration this describes the situation where we ask for a bundle that sits over the total space of another (principal) bundle to descend down along that bundle map to . So beware of the two different roles that bundles play here.
We do not assume to be trivial. We have also the two projections
out of the pullback, where make a kernel pair of . Thus the principality condition is equivalent to saying that make also a kernel pair of its own coequalizer. The two diagrams above are truncations of augmented simplicial objects in . We want to relate these objects to monads.
We now describe the monadic descent along the morphism from above for the codomain fibration .
There are two monads acting on the overcategory whose underlying functors are
The first monad, is the usual one for monadic descent along induced from a pair of adjoint functors.
The second one, , exists due to the principality of and is defined as follows:
to construct the component of the transformation where , by the universal property of the pullback there is an obvious map to
which can be interpreted as a map because the domains of the maps and are the same by the definition and the commuting triangles can be checked easily.
This rule extends to an isomorphism of monads
As a corollary, the Eilenberg-Moore categories of the two monads are equivalent. Notice that the actions over the monad are certain maps , hence by adjointness. This matches one of the definitions for an equivariant sheaf.
The map of the principal bundle is an effective descent morphism with respect to the codomain fibration if the comparison functor for any of the two above isomorphic monads above is an equivalence of categories.
There is a bifibration from the category of modules over any ring, mapping each module to the ring that it is a module over. This models, dually, an algebraic version of vector bundles over affine schemes.
Comonadic descent for this bifibration (equivalently monadic descent for its formal dual, ) is the same as descent for a Sweedler coring. See there for details and geometric interpretations.
Another example is in gluing categories from localizations.
All the ingredients of monadic descent generalize from category theory to higher category theory. Accordingly, one may consider higher monadic descent that relates to ∞-stacks as monadic descent relates to stacks. For more on this see
A review is in
The Bénabou-Roubaud theorem on monadic descent is due to