nLab
monadic descent

Idea

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 $F⊣U$ where

$$F:E_{b_1}=:A\leftrightarrow B:=E_{b_2}:U$$F \;:\; E_{b_1} =: A\leftrightarrow B := E_{b_2} \;:\; U

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 $\mathrm{Mod}\to \mathrm{Ring}$ of modules over Rings.

Details

The adjunction, $F⊣U$ with unit $\eta :1_B\to U\circ F$ and counit $ϵ:F\circ U\to 1_A$, induces the monad $T=(T,\mu ,\eta )$ in $B$ via $T≔UF$, $\mu ≔UϵF:UFUF\to UF$ with unit $\eta$. The monad $T$ generates the Eilenberg–Moore category $B^T$ of modules (= algebras) over $T$, that is the pairs $(M,\nu )$ where $M\in \mathrm{Ob}(B)$ and $\nu :TM\to M$ satisfies the action axioms $T(\nu )\circ \nu ={\nu }_T\circ \nu$ and $\nu \circ {\eta }_M={\mathrm{id}}_M$.

We think of this data as follows:

• $A$ is the category of some type of objects on some space;

• $B$ is the category of this type of object on a cover of that space

• $B^T$ is the category of descent data for such objects, with respect to that cover.

There is a comparison functor

$$K:A\to B^T,K(N)≔(U(N),U({ϵ}_N)).$$K : A\to B^{\mathbf{T}}, \,\,\,\, K(N) \coloneqq (U(N),U(\epsilon_N)).

We say that the functor $U$ is monadic or more informally that $A$ is monadic over $B$ if the comparison functor is an equivalence of categories. Sometimes one is interested in a weaker condition: whether the comparison functor is fully faithful.

Main theorems

The main theorem is Beck’s monadicity theorem.

Given a Grothendieck bifibration $p:E\to B$ and a morphism $f:b\to b\prime$ in the base category $B$, one can choose a direct image $f_{!}:E_b\to E_{b\prime }$ and an inverse image functor $f^{*}:E_{b\prime }\to E_b$, which form an adjunction $f_{!}⊣f^{*}$. 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 $f_{!}⊣f^{*}$ 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.

Examples

Monadic descent of bundles

One of the most basic examples of bifibrations are codomain fibrations $\mathrm{cod}:[I,C]\to C$. 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 $\mathrm{cod}:[I,C]\to C$ 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.

Motivation: failure of push-forward for principal bundles

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 $P\to Z$ liveing 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 $\mathrm{Top}/Z$

• 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

$$\mathrm{push}(f):\mathrm{Top}/X\to \mathrm{Top}/Z$$push(f) : Top/X \to Top/Z

followed by pullback

$$\mathrm{pull}(f):\mathrm{Top}/Z\to \mathrm{Top}/X$$pull(f) : Top/Z \to Top/X

is nothing but the monad associated to $f:X\to Z$ with respect to the codomain bifibration $\mathrm{cod}:[I,\mathrm{Top}]\to \mathrm{Top}$.

So by regarding principal bundles $P\to X$ more generally as just objects in the overcategory $\mathrm{Top}/X$ we make the tools of monadic descent applicable to them.

The monad

Let $C$ be a category with pullbacks. Then the codomain fibration

$$\mathrm{cod}:[I,C]\to C$$cod : [I,C] \to C

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 $\mathrm{push}(f)$ associated to a morphism $\pi :Y\to X$ in $C$ is the functor

$$\mathrm{push}(\pi ):C/Y\to C/X$$push(\pi) : C/Y \to C/X

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 $\mathrm{pull}(f)$ associated to $f$ is the functor

$$C/Y\leftarrow C/X:\mathrm{pull}(\pi )$$C/Y \leftarrow C/X : pull(\pi)

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

$$T_{\pi }=\mathrm{pull}(\pi )\circ \mathrm{push}(\pi ):C/Y\to C/Y$$T_\pi = pull(\pi) \circ push(\pi) : C/Y \to C/Y

for the monad built from these two adjoint functors.

The algebras over the monad: geometric descent data

We spell out in detail the data of an algebra over the above monad, nad show that this encodes precisely the familiar geometric descent datum for a bundle.

To that end, let $(P,\rho )$

$$P:*\to C/Y\begin{array}{ccc}& & C/Y\\ & {}^P\nearrow & {\Downarrow }^{\rho }& {\searrow }^T\\ *& & \stackrel{P}{\to }& & C/Y\end{array}$$P : {*} \to C/Y \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \array{ && C/Y \\ & {}^{\mathllap{P}} \nearrow &\Downarrow^{\rho}& \searrow^{\mathrlap{T}} \\ {*} &&\stackrel{P}{\to}&& C/Y }

be an algebra over our monad. In components this is an object $TP$ equipped with a morphism ${\rho }_P:TP\to P$.

The object $TP\in [I,C{]}_Y$ 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

  $$\begin{array}{ccc}Y{\times }_{X}P& \to & P\\ \downarrow & & \downarrow \\ & & Y\\ \downarrow & & {\downarrow }^{\pi }\\ Y& \stackrel{\pi }{\to }& X\end{array}.$$\array{ Y \times_X P &\to& P \\ \downarrow && \downarrow \\ && Y \\ \downarrow && \downarrow^{\mathrlap{\pi}} \\ Y &\stackrel{\pi}{\to}& X } \,.

  This pullback along a composite of morphisms may be computed as two consecutive pullbacks. The first one is

  $$\begin{array}{ccc}Y{\times }_{X}Y& \to & Y\\ \downarrow & & {\downarrow }^{\pi }\\ Y& \stackrel{\pi }{\to }& X\end{array}$$\array{ Y \times_X Y &\to& Y \\ \downarrow && \downarrow^{\mathrlap{\pi}} \\ Y &\stackrel{\pi}{\to}& X }

  which is the first term in the Čech nerve of $\pi$. So the total pullback is the pullback $P$ to $Y{\times }_{X}Y$:

  $$\begin{array}{ccc}(Y{\times }_{X}Y){\times }_{Y}P& \to & P\\ \downarrow & & \downarrow \\ Y{\times }_{X}Y& \to & Y\\ \downarrow & & {\downarrow }^{\pi }\\ Y& \stackrel{\pi }{\to }& X\end{array}.$$\array{ (Y \times_X Y) \times_Y P &\to& P \\ \downarrow && \downarrow \\ Y \times_X Y &\to& Y \\ \downarrow && \downarrow^{\mathrlap{\pi}} \\ Y &\stackrel{\pi}{\to}& X } \,.

Therefore the action ${\rho }_T:TP\to P$ of our monad on $P$ is given in $C$ by a morphism

$$\begin{array}{ccccc}(Y{\times }_{X}Y){\times }_{Y}P& & \stackrel{\rho }{\to }& & P\\ & \searrow & & \swarrow \\ & & Y\end{array}.$$\array{ (Y \times_X Y) \times_Y P &&\stackrel{\rho}{\to}&& P \\ & \searrow && \swarrow \\ && Y } \,.

As an example, think of this in the context $C=\mathrm{Top}$ with $\pi Y\to X$ 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 $Y{\times }_{X}Y={\coprod }_{ij}{U}_i\cap U_j$ is the union of double intersection of covering patches, and $(Y{\times }_{X}Y){\times }_{Y}P=({\coprod }_{ij}{U}_i\cap U_j\times G)$ 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

$$\rho :\coprod _{ij}:(U_i\cap U_j\times G)\to \coprod _i{U}_i\times G$$\rho : \coprod_{i j} : (U_i \cap U_j \times G) \to \coprod_i U_i \times G

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.

Monadic descent along principal bundles

Idea

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.

Setup

Let $C$ be a category with pullbacks and et $G$ 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 $(p,g)\mapsto (p,\mathrm{pg})$ is an isomorphism.

$$P\times G\stackrel{\nu }{\underset{p}{\rightrightarrows }}P\stackrel{\pi }{\to }X$$P\times G \overset{\nu}\underset{p}\rightrightarrows P \overset{\pi}\to X

We do not assume $P$ to be trivial. We have also the two projections

$$P{\times }_{X}P\stackrel{p_1}{\underset{p_2}{\rightrightarrows }}P\stackrel{\pi }{\to }X$$P\times_X P \overset{p_1}\underset{p_2}\rightrightarrows P \overset{\pi}\to X

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.

The two different monads

We now describe the monadic descent along the morphism $\pi :P\to X$ from above for the codomain fibration $\mathrm{cod}:[I,C]\to C$.

There are two monads acting on the overcategory $C/P$ whose underlying functors are

1. $T:={\pi }^{*}{\pi }_{!}$.

2. $\tilde{T}:=p_{!}{\nu }^{*}$

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 $p_{!}{\nu }^{*}h$

$$\begin{array}{c}{\nu }^{*}{p}_{!}{\nu }^{*}L\\ & {\searrow }^{{\mu }_h}\\ & & {\nu }^{*}L& \to & L\\ & & \downarrow & & \downarrow \\ & & P& \stackrel{\stackrel{p}{\to }}{\underrightarrow{\nu }}& X\end{array},$$\array{ \nu^* p_! \nu^* L \\ & \searrow^{\mathrlap{\mu_h}} \\ &&\nu^* L &\to& L \\ && \downarrow && \downarrow \\ && P &\stackrel{\stackrel{p}{\to}}{\underset{\to}{\nu}}& X } \,,

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

$$p_{!}{\nu }^{*}h\cong {\pi }^{*}{\pi }_{!}h$$p_! \nu^* h \cong \pi^* \pi_! h

natural in $h:L\to P$, read off from the double pullback diagram

$$\begin{array}{ccccc}{p}_{!}{\nu }^{*}L& \stackrel{\simeq }{\to }& {\pi }^{*}{\pi }_{!}L& \to & L\\ \downarrow & & \downarrow & & {\downarrow }^h\\ P\times G& \stackrel{\simeq }{\to }& P{\times }_{X}P& \to & P\\ & & \downarrow & & {\downarrow }^{\pi }\\ & & P& \to & X\end{array}.$$\array{ p_! \nu^* L &\stackrel{\simeq}{\to}& \pi^* \pi_! L &\to& L \\ \downarrow && \downarrow && \downarrow^{\mathrlap{h}} \\ P \times G &\stackrel{\simeq}{\to}& P \times_X P &\to& P \\ && \downarrow && \downarrow^{\mathrlap{\pi}} \\ && P &\to& X } \,.

This rule extends to an isomorphism of monads

$$T\simeq \tilde{T}.$$T \simeq \tilde T \,.

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.

Monadic descent of modules

There is a bifibration $\mathrm{Mod}\to \mathrm{Rings}$ 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, ${\mathrm{Mod}}^{\mathrm{op}}\to {\mathrm{Rings}}^{\mathrm{op}}$) is the same as descent for a Sweedler coring. See there for details and geometric interpretations.

Gluing categories from localizations

Another example is in gluing categories from localizations.

Higher category theoretical version

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

• higher monadic descent.

Related concepts

• descent

  • cover

  • cohomological descent

  • descent morphism

  • monadic descent

    • Sweedler coring

    • higher monadic descent

    • descent in noncommutative algebraic geometry

• sheaf, (2,1)-sheaf, 2-sheaf (∞,1)-sheaf

References

The Bénabou-Roubaud theorem on monadic descent is due to

• Jean Bénabou, Jacques Roubaud, Monades et descente, C. R. Acad. Sc. Paris, t. 270 (12 Janvier 1970), Serie A, 96–98, (link, Bibliothèque nationale de France)