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+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Locality and descent +--{: .hide} [[!include descent and locality - contents]] =-- #### Cohomology +--{: .hide} [[!include cohomology - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea Monadic descent is a way to encode [[descent]] of [[fibered category|fibered categories]] (equivalently, by the [[Grothendieck construction]], of [[pseudofunctor]]s) that have the special property that they are [[bifibration|bifibrations]]. This allows the use of algebraic tools, notably [[monad|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-[[monad|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 $$ F \;:\; E_{b_1} =: A\leftrightarrow B := E_{b_2} \;:\; U $$ 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 $Mod \to Ring$ of [[module]]s over [[Ring]]s. ## Definition {#Definition} 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 $$ (f_! \dashv f^\ast \dashv f_\ast) \colon \mathcal{C}_X \stackrel{\overset{f_!}{\longrightarrow}}{\stackrel{\overset{f^\ast}{\longleftarrow}}{\underset{f_\ast}{\longrightarrow}}} \mathcal{C}_Y $$ for the corresponding [[base change]] [[adjoint triple]], and write $$ (T_f \dashv J_f) \coloneqq (f^\ast f_! \dashv f^\ast f_\ast) $$ 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 $$ \mathcal{C}_{Y} \longrightarrow Desc_{\mathcal{C}}(f) \,. $$ 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 $$ Desc_{\mathcal{C}}(f) \simeq EM(T_f) \,. $$ 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)](#JanelidzeTholen94)). ## Properties 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 category|Eilenberg–Moore categories]] of monads, or dually comonads. ## Examples ### Descent for the codomain fibration {#DescentForTheCodomainFibration} 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](#JanelidzeTholen94)). Hence for $f \colon X \longrightarrow Y$ any morphism in $\mathcal{C}$ and $$ (f_! \dashv f^\ast \dashv f_\ast) \colon \mathcal{C}_{/X} \longrightarrow \mathcal{C}_{/Y} $$ 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](conservative+functor#PullbackAlongEpimorphisms) in the [[monadicity theorem]].) ### Monadic descent of bundles {#ForCodomainFibs} One of the most basic examples of [[bifibration]]s are [[codomain fibration]]s $cod : [I,C] \to C$, where $[I,C]$ is the [[arrow category]] of $C$ and $cod$ sends any arrow in $C$ to its [[codomain]]. Accordingly, monadic descent applied to codomain fibrations archetypically exhibits the nature of monadic descent. We therefore spell out this example in 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 coring]]s, 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. #### 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 bundle]]s, but that can also be pushed forward, such as [[vector bundle]]s (with suitable care taken) or more generally [[module]]s 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 [[fiber]]s 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 $$ push(f) : Top/X \to Top/Z $$ followed by pullback $$ pull(f) : Top/Z \to Top/X $$ is nothing but the [[monad]] associated to $f : X \to Z$ with respect to the [[codomain fibration|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. #### The monad Let $C$ be a [[category]] with [[pullback]]s. Then the [[codomain fibration]] $$ cod : [I,C] \to C $$ 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 $push(f)$ associated to a morphism $\pi : Y \to X$ in $C$ is the functor $$ push(\pi) : C/Y \to C/X $$ 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 $pull(f)$ associated to $f$ is the functor $$ C/Y \leftarrow C/X : pull(\pi) $$ 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 $$ T_\pi = pull(\pi) \circ push(\pi) : C/Y \to C/Y $$ for the [[monad]] built from these two [[adjoint functor]]s. #### 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 $(P, \rho)$ $$ P : {*} \to C/Y \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \array{ && C/Y \\ & {}^{\mathllap{P}} \nearrow &\Downarrow^{\rho}& \searrow^{\mathrlap{T}} \\ {*} &&\stackrel{P}{\to}&& C/Y } $$ be an [[monad|algebra over]] our monad. In components this is an object $P$ 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 $$ \array{ Y \times_X P &\to& P \\ \downarrow && \downarrow \\ && Y \\ \downarrow && \downarrow^{\mathrlap{\pi}} \\ Y &\stackrel{\pi}{\to}& X } \,. $$ This [[pullback]] along a composite of morphisms may be computed as two consecutive pullbacks. The first one is $$ \array{ Y \times_X Y &\to& Y \\ \downarrow && \downarrow^{\mathrlap{\pi}} \\ Y &\stackrel{\pi}{\to}& X } $$ which is the first term in the [[Čech nerve]] of $\pi$. So the total pullback is the pullback $P$ to $Y\times_X Y$: $$ \array{ (Y \times_X Y) \times_Y P &\to& P \\ \downarrow && \downarrow \\ Y \times_X Y &\to& Y \\ \downarrow && \downarrow^{\mathrlap{\pi}} \\ Y &\stackrel{\pi}{\to}& X } \,. $$ Therefore the action $\rho_T : T P \to P$ of our [[monad]] on $P$ is given in $C$ by a morphism $$ \array{ (Y \times_X Y) \times_Y P &&\stackrel{\rho}{\to}&& P \\ & \searrow && \swarrow \\ && Y } \,. $$ As an example, think of this in the context $C = Top$ 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 $$ \rho : \coprod_{i j} : (U_i \cap U_j \times G) \to \coprod_i U_i \times G $$ 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 {#AlongPrincipalBundle} #### Idea In the [above section](#ForCodomainFibs) we considered monadic descent _of_ [[bundle]]s $P \to Y$ _along_ morphisms $f : Y \to X$. Now we consider monadic descent _along_ morphisms $f : P \to X$ that happen to be $G$-[[principal bundle]]s, 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 [[pullback]]s 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 [[principal bundle|principality condition]] says that $P\times G \to P\times_X P$ given by $(p,g)\mapsto (p,pg)$ is an [[isomorphism]]. $$ P\times G \overset{\nu}\underset{p}\rightrightarrows P \overset{\pi}\to X $$ We do not assume $P$ to be trivial. We have also the two projections $$ P\times_X P \overset{p_1}\underset{p_2}\rightrightarrows P \overset{\pi}\to X $$ 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 object]]s in $C$. We want to relate these objects to [[monad]]s. #### The two different 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 1. $T := \pi^* \pi_!$. 1. $\tilde T := p_! \nu^*$ The first monad, $T$ is the usual one for monadic descent along $\pi$ induced from a pair of [[adjoint functor]]s. 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 $p_! \nu^* h$ $$ \array{ \nu^* p_! \nu^* L \\ & \searrow^{\mathrlap{\mu_h}} \\ &&\nu^* L &\to& L \\ && \downarrow && \downarrow \\ && P &\stackrel{\stackrel{p}{\to}}{\underset{\to}{\nu}}& X } \,, $$ 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 [[principal bundle|principality]] $P\times G \cong P\times_X P$ now induces the [[isomorphism]] $$p_! \nu^* h \cong \pi^* \pi_! h$$ [[natural isomorphism|natural]] in $h:L\to P$, read off from the double [[pullback]] diagram $$ \array{ p_! \nu^* L &\stackrel{\simeq}{\to}& \pi^* \pi_! L &\to& L \\ \downarrow && \downarrow && \downarrow^{\mathrlap{h}} \\ P \times G &\stackrel{\simeq}{\to}& P \times_X P &\to& P \\ && \downarrow && \downarrow^{\mathrlap{\pi}} \\ && P &\to& X } \,. $$ This rule extends to an isomorphism of monads $$ T \simeq \tilde T \,. $$ As a corollary, the [[Eilenberg–Moore category|Eilenberg–Moore categories]] of the two monads are [[equivalence of categories|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]] $Mod \to Rings$ 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 bundle]]s over [[affine scheme]]s. 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. ### Gluing categories from localizations Another example is in [[zoranskoda:gluing categories from localizations|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 [[∞-stack]]s as monadic descent relates to [[stack]]s. For more on this see * [[higher monadic descent]]. ## Related concepts * [[surjective geometric morphism]] * [[descent]] * [[cover]] * [[cohomological descent]] * [[descent morphism]] * **monadic descent** * [[Beck-Chevalley condition]] * [[monadicity theorem]] * [[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 * {#BenabouRoubaud70} [[Jean Bénabou]], [[Jacques Roubaud]], _Monades et descente_, C. R. Acad. Sc. Paris, Ser. A **270** (1970) 96-98 [[gallica:12148/bpt6k480298g/f100](http://gallica.bnf.fr/ark:/12148/bpt6k480298g/f100), [[BenabouRoubaud-MonadesEtDescente.pdf:file]], English translation (by [[Peter Heinig]]): [MO:q/279152](https://mathoverflow.net/q/279152)] Review and further developments: * {#JanelidzeTholen94} [[George Janelidze]], [[Walter Tholen]], *Facets of descent I*, Applied Categorical Structures **2** 3 (1994) 245-281 [[doi:10.1007/BF00878100](https://doi.org/10.1007/BF00878100)] * {#JanelidzeTholen97} [[George Janelidze]], [[Walter Tholen]], *Facets of descent II*, Applied Categorical Structures **5** 3 (1997) 229-248 [[doi:10.1023/A:1008697013769](https://doi.org/10.1023/A:1008697013769)] * [[George Janelidze]], Manuela Sobral, [[Walter Tholen]], _Beyond Barr exactness: effective descent morphisms_, Ch. 8 of Categ. Foundations, (eds. Maria Cristina Pedicchio, Walter Tholen) Enc. Math. Appl. 2003 * Bachuki Mesablishvili, _Monads of effective descent type and comonadicity_, Theory and Applications of Categories __16__:1 (2006) 1-45, [link](http://tac.mta.ca/tac/volumes/16/1/16-01abs.html); _Pure morphisms of commutative rings are effective descent morphisms for modules—a new proof_, Theory Appl. Categ. 7(3), 38–42 (2000) * [[Francis Borceux]], Stefan Caenepeel, [[George Janelidze]], _Monadic approach to Galois descent and cohomology_, [arXiv:0812.1674](http://arxiv.org/abs/0812.1674) * S. Caenepeel, _Galois corings from the descent theory point of view,_ in: Fields Inst. Commun. __43__ (2004) 163–186 * [[Tomasz Brzeziński]], Adrian Vazquez Marquez, Joost Vercruysse, _The Eilenberg–Moore category and a Beck-type theorem for a Morita context_, J. Appl Categor Struct (2011) 19: 821 [doi](https://doi.org/10.1007/s10485-009-9217-0) In triangulated setup there are several results including * P. Balmer, _Descent in triangulated categories_, Mathematische Annalen (2012) 353:1, 109–125 Discussion in [[homotopy theory]] for [[(infinity,1)-monads]] is in * {#Hess10} [[Kathryn Hess]], section 2 of _A general framework for homotopic descent and codescent_ ([arXiv:1001.1556](http://arxiv.org/abs/1001.1556)) [[!redirects comonadic descent]]