group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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 $E \to B$ 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 $f : b_1 \to b_2$ in the base $B$ induces an adjunction $F\dashv U$ where
and we ask whether $U$ is a monadic functor.
This is the original description of descent of presheaves with values in 1-categories due to Alexander Grothendieck.
The archetypical and motivating example is that of the bifibration $Mod \to Ring$ of modules over Rings.
Let $\mathcal{C}$ be a category and $\mathcal{C}_{(-)}$ a bifibration over it. For $f \colon X \longrightarrow Y$ a morphism in $\mathcal{C}$ write
for the corresponding base change adjoint triple, and write
for the induced adjoint pair of a monad $T$ and a comonad $J$.
There is a standard definition of a category $Desc_{\mathcal{C}}(f)$ of descent data for $\mathcal{C}_{(-)}$ along $f$, which comes with a canonical morphism
The morphism $f$ is called (with respect to the given bifibration $\mathcal{C}_{(-)}$)
a descent morphism if this comparison map is a full and faithful functor;
an effective descent morphism if the comparison map is an equivalence of categories.
Now the Bénabou–Roubaud theorem states that if $\mathcal{C}_{(-)}$ satisfies the Beck-Chevalley condition, then descent data is equivalent to the structure of an algebra over a monad for $T_f$ (equivalently a coalgebra for $J_f$), hence is the Eilenberg-Moore category for these (co-)monads
Therefore when $\mathcal{C}_{(-)}$ satisfies the BC condition, then a morphism $f$ is effective descent precisely if $f^\ast \colon \mathcal{C}_{Y} \to \mathcal{C}_{X}$ is monadic, and is descent precisely if $f^\ast$ is of descent type.
This is the monadic formulation of descent theory, “monadic descent”.
(e.g. Janelidze-Tholen 94, pp. 247-248 (3-4 of 37)).
The main theorem is Beck’s monadicity theorem.
Given a Grothendieck bifibration $p:E\to B$ and a morphism $f:b\to b'$ in the base category $B$, one can choose a direct image $f_!:E_b\to E_{b'}$ and an inverse image functor $f^*:E_{b'}\to E_b$, which form an adjunction $f_!\dashv f^*$. Under some conditions (see the Bénabou–Roubaud theorem), the morphism $f$ is an effective descent morphism (with respect to $p$ as a fibered category) iff the comparison functor for the monad induced by the adjunction $f_!\dashv f^*$ is monadic.
We should now see that some instances of categories of descent data are canonically equivalent to and can be reexpressed via Eilenberg–Moore categories of monads, or dually comonads.
Let $\mathcal{C}$ be a locally Cartesian closed category with coequalizers (e.g. a topos). Then effective descent morphisms for the codomain fibration are precisely the regular epimorphisms. (Janelidze-Tholen 94, 2.4).
Hence for $f \colon X \longrightarrow Y$ any morphism in $\mathcal{C}$ and
the induced base change adjoint triple, then $\mathcal{C}_{/Y}$ is equivalent to the Eilenberg-Moore category of algebras over $f^\ast f_!$ (equivalently: of coalgebras of $f^\ast f_\ast$) precisely if $f$ is an effective epimorphism.
(Use conservative pullback along epimorphisms in the monadicity theorem.)
One of the most basic examples of bifibrations are codomain fibrations $cod : [I,C] \to C$. 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 $Y \in C$ is a morphism $P \to Y$, hence a bundle in $C$, 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).
We show in the following that for $cod : [I,C] \to C$ a codomain fibration and for $\pi : Y\to X$ a morphism in $C$, an algebra object in $[I,C]_Y$ over the monad $f^* f_*$ encodes and is encoded by a “geometric” descent datum: that it is
a morphism $P \to Y$
equipped with a transition function between its two pullbacks to double $Y \times_X Y$
which satisfies on $Y \times_X Y \times_X Y$ 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 $P \to X$ (a topological one, say, i.e. a morphism in Top) and a morphism of base spaces $f : X \to Z$, the would-be candidate for the push-forward of $P$ along $f$ is simply the composite map $P \to X \to Z$, regarded as a total space $P \to Z$ living over $Z$.
While that always exists as such, it will in general not be a principal bundle anymore: the fibers of $P \to Z$ over points $z \in Z$ consist of many copies of the fibers of $P \to X$ over points in $X$. 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 $P \to X$ and
first push it forward in this generalized sense to an object $P \to Z$ in the overcategory $Top/Z$
and then pull back the result of that again along $X \to Z$ the result, while still not a principal bundle, is the total space $P$ of the bundle pulled back to the first term in the Čech nerve of $f : X \to Z$. 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
is nothing but the monad associated to $f : X \to Z$ with respect to the codomain bifibration $cod : [I,Top] \to Top$.
So by regarding principal bundles $P \to X$ more generally as just objects in the overcategory $Top/X$ we make the tools of monadic descent applicable to them.
Let $C$ be a category with pullbacks. Then the codomain fibration
is a bifibration (as described there, in detail). Its fiber over an object $X \in C$ is the overcategory $C/X$.
The direct image operation $push(f)$ associated to a morphism $\pi : Y \to X$ in $C$ is the functor
obtained by postcomposition with $f$, which sends $(P \to Y) \in C/Y$ to the composite $P \to Y \stackrel{\pi}{\to} X$ in $C$, regarded as an object of $C/X$.
The inverse image operation $pull(f)$ associated to $f$ is the functor
obtained by pullback in $C$ along $\pi$, which sends $(Q \to X) \in C/X$ the pullback $Q \times_X Y$, regarded as an object of $C/Y$ in terms of the canonical projection morphism $Q \times_X Y\to Y$ out of the pullback.
Write
for the monad built from these two adjoint functors.
We spell out in detail the data of an algebra over the above monad, and show that this encodes precisely the familiar geometric descent datum for a bundle.
To that end, let $(P, \rho)$
be an algebra over our monad. In components this is an object $T P$ equipped with a morphism $\rho_P : T P \to P$.
The object $T P \in [I,C]_Y$ is given by
first pushing $P \to Y$ forward along $\pi : Y \to X$ to the object $P \to Y \to X$.
then pulling this back along $\pi$ 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 $\pi$. So the total pullback is the pullback $P$ to $Y\times_X Y$:
Therefore the action $\rho_T : T P \to P$ of our monad on $P$ is given in $C$ by a morphism
As an example, think of this in the context $C = Top$ with $\pi \colon Y \to X$ coming from an open cover $\{U_i \to X\}$ of $X$ with $Y = \coprod_i U_i$, and with $P = Y \times G$ a trivial $G$-principal bundle for some group $G$. Then the space $Y \times_X Y = \coprod_{i j} U_i \cap U_j$ is the union of double intersection of covering patches, and $(Y \times_X Y) \times_Y P = (\coprod_{i j} U_i \cap U_j \times G)$ is to be thought of as the trivial $G$-principal bundle over $U_j$, restricted to the intersection. In this case our morphism $\rho$ acts as
and thus maps the trivial $G$-bundle over $U_j$ on the intersection with the trivial $G$-bundle over $U_i$. So it is a transition function. If this is a $G$-equivariant, it may be part of the descent datum for the $G$-principal bundle.
In the above section we considered monadic descent of bundles $P \to Y$ along morphisms $f : Y \to X$.
Now we consider monadic descent along morphisms $f : P \to X$ that happen to be $G$-principal bundles, for some group object $G$. When considered with respect to the codomain fibration this describes the situation where we ask for a bundle $L \to P$ that sits over the total space of another (principal) bundle to descend down along that bundle map to $X$. So beware of the two different roles that bundles play here.
Let $C$ be a category with pullbacks and let $G$ be an internal group in $C$.
Let $\nu: P\times G\to P$ be a right principal action and $p:P\times G\to P$ the projection. Let $\pi:P\to X$ be the coequalizer of $\nu$ and $p$. The principality condition says that $P\times G \to P\times_X P$ given by $(p,g)\mapsto (p,pg)$ is an isomorphism.
We do not assume $P$ to be trivial. We have also the two projections
out of the pullback, where $p_1,p_2$ make a kernel pair of $\pi$. Thus the principality condition is equivalent to saying that $\nu,p$ make also a kernel pair of its own coequalizer. The two diagrams above are truncations of augmented simplicial objects in $C$. We want to relate these objects to monads.
We now describe the monadic descent along the morphism $\pi : P \to X$ from above for the codomain fibration $cod : [I,C] \to C$.
There are two monads acting on the overcategory $C/P$ whose underlying functors are
$T := \pi^* \pi_!$.
$\tilde T := p_! \nu^*$
The first monad, $T$ is the usual one for monadic descent along $\pi$ induced from a pair of adjoint functors.
The second one, $\tilde T$, exists due to the principality of $P \to X$ and is defined as follows:
to construct the component $\mu_h$ of the transformation $\mu: p_! \nu^* p_!\nu^*\to p_!\nu^*$ where $h: L\to P$, by the universal property of the pullback there is an obvious map $\nu^* p_! \nu^* h$ to $p_! \nu^* h$
which can be interpreted as a map $p_!\nu^* p_! \nu^* h\to p_* \nu^* h$ because the domains of the maps $p_!\nu^* p_! \nu^* h$ and $\nu^* p_! \nu^* h$ are the same by the definition and the commuting triangles can be checked easily.
The principality $P\times G \cong P\times_X P$ now induces the isomorphism
natural in $h:L\to P$, read off from the double pullback diagram
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 $p_! \nu^*$ are certain maps $p_!\nu^*h\to h$, hence $\nu^* h\to p^* h$ by adjointness. This matches one of the definitions for an equivariant sheaf.
The map $\pi : P\to X$ 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 $Mod \to Rings$ 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, $Mod^{op} \to Rings^{op}$) 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
The Bénabou-Roubaud theorem on monadic descent is due to
Review and further developments include
George Janelidze, Walter Tholen, Facets of Descent I, Applied Categorical Structures 1994, Volume 2, Issue 3, pp 245-281
George Janelidze, Walter Tholen, Facets of Descent II, Applied Categorical Structures September 1997, Volume 5, Issue 3, pp 229-248
Francis Borceux, S. Caenepeel, George Janelidze, Monadic approach to Galois descent and cohomology (arXiv:0812.1674)
Discussion in homotopy theory for (infinity,1)-monads is in