monadic descent


Locality and descent



Special and general types

Special notions


Extra structure





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 EBE \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 1b 2f : b_1 \to b_2 in the base BB induces an adjunction FUF\dashv U where

F:E b 1=:AB:=E b 2:U F \;:\; E_{b_1} =: A\leftrightarrow B := E_{b_2} \;:\; U

and we ask whether UU 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 ModRingMod \to Ring of modules over Rings.


Let 𝒞\mathcal{C} be a category and 𝒞 ()\mathcal{C}_{(-)} a bifibration over it. For f:XYf \colon X \longrightarrow Y a morphism in 𝒞\mathcal{C} write

(f !f *f *):𝒞 Xf *f *f !𝒞 Y (f_! \dashv f^\ast \dashv f_\ast) \colon \mathcal{C}_X \stackrel{\overset{f_!}{\longrightarrow}}{\stackrel{\overset{f^\ast}{\longleftarrow}}{\underset{f_\ast}{\longrightarrow}}} \mathcal{C}_Y

for the corresponding base change adjoint triple, and write

(T fJ f)(f *f !f *f *) (T_f \dashv J_f) \coloneqq (f^\ast f_! \dashv f^\ast f_\ast)

for the induced adjoint pair of a monad TT and a comonad JJ.

There is a standard definition of a category Desc 𝒞(f)Desc_{\mathcal{C}}(f) of descent data for 𝒞 ()\mathcal{C}_{(-)} along ff, which comes with a canonical morphism

𝒞 YDesc 𝒞(f). \mathcal{C}_{Y} \longrightarrow Desc_{\mathcal{C}}(f) \,.

The morphism ff is called (with respect to the given bifibration 𝒞 ()\mathcal{C}_{(-)})

Now the Bénabou–Roubaud theorem states that if 𝒞 ()\mathcal{C}_{(-)} satisfies the Beck-Chevalley condition, then descent data is equivalent to the structure of an algebra over a monad for T fT_f (equivalently a coalgebra for J fJ_f), hence is the Eilenberg-Moore category for these (co-)monads

Desc 𝒞(f)EM(T f). Desc_{\mathcal{C}}(f) \simeq EM(T_f) \,.

Therefore when 𝒞 ()\mathcal{C}_{(-)} satisfies the BC condition, then a morphism ff is effective descent precisely if f *:𝒞 Y𝒞 Xf^\ast \colon \mathcal{C}_{Y} \to \mathcal{C}_{X} is monadic, and is descent precisely if f *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:EBp:E\to B and a morphism f:bbf:b\to b' in the base category BB, one can choose a direct image f !:E bE bf_!:E_b\to E_{b'} and an inverse image functor f *:E bE bf^*:E_{b'}\to E_b, which form an adjunction f !f *f_!\dashv f^*. Under some conditions (see the Bénabou–Roubaud theorem), the morphism ff is an effective descent morphism (with respect to pp as a fibered category) iff the comparison functor for the monad induced by the adjunction f !f *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.


Descent for the codomain fibration

Let 𝒞\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:XYf \colon X \longrightarrow Y any morphism in 𝒞\mathcal{C} and

(f !f *f *):𝒞 /X𝒞 /Y (f_! \dashv f^\ast \dashv f_\ast) \colon \mathcal{C}_{/X} \longrightarrow \mathcal{C}_{/Y}

the induced base change adjoint triple, then 𝒞 /Y\mathcal{C}_{/Y} is equivalent to the Eilenberg-Moore category of algebras over f *f !f^\ast f_! (equivalently: of coalgebras of f *f *f^\ast f_\ast) precisely if ff is an effective epimorphism.

(Use conservative pullback along epimorphisms in the monadicity theorem.)

Monadic descent of bundles

One of the most basic examples of bifibrations are codomain fibrations cod:[I,C]Ccod : [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 YCY \in C is a morphism PYP \to Y, hence a bundle in CC, 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]Ccod : [I,C] \to C a codomain fibration and for π:YX\pi : Y\to X a morphism in CC, an algebra object in [I,C] Y[I,C]_Y over the monad f *f *f^* f_* encodes and is encoded by a “geometric” descent datum: that it is

  • a morphism PYP \to Y

  • equipped with a transition function between its two pullbacks to double Y× XYY \times_X Y

  • which satisfies on Y× XY× XYY \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 PXP \to X (a topological one, say, i.e. a morphism in Top) and a morphism of base spaces f:XZf : X \to Z, the would-be candidate for the push-forward of PP along ff is simply the composite map PXZP \to X \to Z, regarded as a total space PZP \to Z living over ZZ.

While that always exists as such, it will in general not be a principal bundle anymore: the fibers of PZP \to Z over points zZz \in Z consist of many copies of the fibers of PXP \to X over points in XX. 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 PXP \to X and

  • first push it forward in this generalized sense to an object PZP \to Z in the overcategory Top/ZTop/Z

  • and then pull back the result of that again along XZX \to Z the result, while still not a principal bundle, is the total space PP of the bundle pulled back to the first term in the Čech nerve of f:XZf : 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

push(f):Top/XTop/Z push(f) : Top/X \to Top/Z

followed by pullback

pull(f):Top/ZTop/X pull(f) : Top/Z \to Top/X

is nothing but the monad associated to f:XZf : X \to Z with respect to the codomain bifibration cod:[I,Top]Topcod : [I,Top] \to Top.

So by regarding principal bundles PXP \to X more generally as just objects in the overcategory Top/XTop/X we make the tools of monadic descent applicable to them.

The monad

Let CC be a category with pullbacks. Then the codomain fibration

cod:[I,C]C cod : [I,C] \to C

is a bifibration (as described there, in detail). Its fiber over an object XCX \in C is the overcategory C/XC/X.

The direct image operation push(f)push(f) associated to a morphism π:YX\pi : Y \to X in CC is the functor

push(π):C/YC/X push(\pi) : C/Y \to C/X

obtained by postcomposition with ff, which sends (PY)C/Y(P \to Y) \in C/Y to the composite PYπXP \to Y \stackrel{\pi}{\to} X in CC, regarded as an object of C/XC/X.

The inverse image operation pull(f)pull(f) associated to ff is the functor

C/YC/X:pull(π) C/Y \leftarrow C/X : pull(\pi)

obtained by pullback in CC along π\pi, which sends (QX)C/X(Q \to X) \in C/X the pullback Q× XYQ \times_X Y, regarded as an object of C/YC/Y in terms of the canonical projection morphism Q× XYYQ \times_X Y\to Y out of the pullback.


T π=pull(π)push(π):C/YC/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, and show that this encodes precisely the familiar geometric descent datum for a bundle.

To that end, let (P,ρ)(P, \rho)

P:*C/Y C/Y P ρ T * P C/Y 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 TPT P equipped with a morphism ρ P:TPP\rho_P : T P \to P.

The object TP[I,C] YT P \in [I,C]_Y is given by

  • first pushing PYP \to Y forward along π:YX\pi : Y \to X to the object PYXP \to Y \to X.

  • then pulling this back along π\pi to yield the left vertical morphism in

    Y× XP P Y π Y π X. \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

    Y× XY Y π Y π X \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 π\pi. So the total pullback is the pullback PP to Y× XYY\times_X Y:

    (Y× XY)× YP P Y× XY Y π Y π X. \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 ρ T:TPP\rho_T : T P \to P of our monad on PP is given in CC by a morphism

(Y× XY)× YP ρ P Y. \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=TopC = Top with π:YX\pi \colon Y \to X coming from an open cover {U iX}\{U_i \to X\} of XX with Y= iU iY = \coprod_i U_i, and with P=Y×GP = Y \times G a trivial GG-principal bundle for some group GG. Then the space Y× XY= ijU iU jY \times_X Y = \coprod_{i j} U_i \cap U_j is the union of double intersection of covering patches, and (Y× XY)× YP=( ijU iU j×G)(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 GG-principal bundle over U jU_j, restricted to the intersection. In this case our morphism ρ\rho acts as

ρ: ij:(U iU j×G) iU i×G \rho : \coprod_{i j} : (U_i \cap U_j \times G) \to \coprod_i U_i \times G

and thus maps the trivial GG-bundle over U jU_j on the intersection with the trivial GG-bundle over U iU_i. So it is a transition function. If this is a GG-equivariant, it may be part of the descent datum for the GG-principal bundle.

Monadic descent along principal bundles


In the above section we considered monadic descent of bundles PYP \to Y along morphisms f:YXf : Y \to X.

Now we consider monadic descent along morphisms f:PXf : P \to X that happen to be GG-principal bundles, for some group object GG. When considered with respect to the codomain fibration this describes the situation where we ask for a bundle LPL \to P that sits over the total space of another (principal) bundle to descend down along that bundle map to XX. So beware of the two different roles that bundles play here.


Let CC be a category with pullbacks and let GG be an internal group in CC.

Let ν:P×GP\nu: P\times G\to P be a right principal action and p:P×GPp:P\times G\to P the projection. Let π:PX\pi:P\to X be the coequalizer of ν\nu and pp. The principality condition says that P×GP× XPP\times G \to P\times_X P given by (p,g)(p,pg)(p,g)\mapsto (p,pg) is an isomorphism.

P×GpνPπX P\times G \overset{\nu}\underset{p}\rightrightarrows P \overset{\pi}\to X

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

P× XPp 2p 1PπX P\times_X P \overset{p_1}\underset{p_2}\rightrightarrows P \overset{\pi}\to X

out of the pullback, where p 1,p 2p_1,p_2 make a kernel pair of π\pi. Thus the principality condition is equivalent to saying that ν,p\nu,p make also a kernel pair of its own coequalizer. The two diagrams above are truncations of augmented simplicial objects in CC. We want to relate these objects to monads.

The two different monads

We now describe the monadic descent along the morphism π:PX\pi : P \to X from above for the codomain fibration cod:[I,C]Ccod : [I,C] \to C.

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

  1. T:=π *π !T := \pi^* \pi_!.

  2. T˜:=p !ν *\tilde T := p_! \nu^*

The first monad, TT is the usual one for monadic descent along π\pi induced from a pair of adjoint functors.

The second one, T˜\tilde T, exists due to the principality of PXP \to X and is defined as follows:

to construct the component μ h\mu_h of the transformation μ:p !ν *p !ν *p !ν *\mu: p_! \nu^* p_!\nu^*\to p_!\nu^* where h:LPh: L\to P, by the universal property of the pullback there is an obvious map ν *p !ν *h\nu^* p_! \nu^* h to p !ν *hp_! \nu^* h

ν *p !ν *L μ h ν *L L P νp X, \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 !ν *p !ν *hp *ν *hp_!\nu^* p_! \nu^* h\to p_* \nu^* h because the domains of the maps p !ν *p !ν *hp_!\nu^* p_! \nu^* h and ν *p !ν *h\nu^* p_! \nu^* h are the same by the definition and the commuting triangles can be checked easily.

The principality P×GP× XPP\times G \cong P\times_X P now induces the isomorphism

p !ν *hπ *π !hp_! \nu^* h \cong \pi^* \pi_! h

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

p !ν *L π *π !L L h P×G P× XP P π P X. \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

TT˜. 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 !ν *p_! \nu^* are certain maps p !ν *hhp_!\nu^*h\to h, hence ν *hp *h\nu^* h\to p^* h by adjointness. This matches one of the definitions for an equivariant sheaf.

The map π:PX \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 ModRingsMod \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 opRings opMod^{op} \to Rings^{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


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)

Review and further developments include

Discussion in homotopy theory for (infinity,1)-monads is in

Revised on June 26, 2017 12:27:15 by Mike Shulman (