nLab
Sweedler coring

Idea

A Sweedler coring is an algebraic structure that is roughly the formal dual of the Čech nerve of a cover: it is used to describe descent in algebraic contexts.

See also monadic descent.

Definition

Let $f:R\hookrightarrow S$ be the extension of associative unital $k$-algebras (where $k$ is a commutative unital ring).

The corresponding canonical coring or Sweedler coring is the $S$-coring

$$C=S{\otimes }_{R}S$$C = S\otimes_R S

with coproduct

$$\Delta :C\to C{\otimes }_{S}C\cong S{\otimes }_{R}S{\otimes }_{R}S$$\Delta : C\to C\otimes_S C \cong S\otimes_R S\otimes_R S

given by

$$\Delta :s_1\otimes s_2\mapsto s_1\otimes 1\otimes s_2$$\Delta: s_1\otimes s_2 \mapsto s_1\otimes 1 \otimes s_2

and counit

$$ϵ:C\to S$$\epsilon : C\to S

given by

$$ϵ:s_1\otimes s_2\mapsto s_1{s}_2.$$\epsilon: s_1 \otimes s_2 \mapsto s_1 s_2 \,.

The element $1\otimes 1$ is a grouplike element in the Sweedler’s coring.

Geometric interpretation

We give a dual geometric interpretation of the Sweedler coring.

Suppose a context of spaces and function algebras on spaces that satisfies the basic axioms of geometric function theory, in that the algebra of functions $C(Y_1{\times }_X{Y}_2)$ on a fiber product

$$\begin{array}{ccc}{Y}_1{\times }_X{Y}_2& \to & Y_2\\ \downarrow & & \downarrow \\ Y_1& \to & X\end{array}$$\array{ Y_1 \times_X Y_2 &\to& Y_2 \\ \downarrow && \downarrow \\ Y_1 &\to& X }

is the tensor product of the functions on the factors:

$$C(Y_1{\times }_X{Y}_2)=C(Y_1){\otimes }_{C(X)}C(Y_2).$$C(Y_1 \times_X Y_2) = C(Y_1) \otimes_{C(X)} C(Y_2) \,.

Then let $\pi :Y\to X$ be a morphism of spaces and set

$$R:=C(X)$$R := C(X)

and

$$S=C(Y)$$S = C(Y)

and

$$(R\hookrightarrow S):={\pi }^{*}:C(X)\to C(Y).$$(R \hookrightarrow S) := \pi^* : C(X) \to C(Y) \,.

The morphism $\pi$ induces its augmented Čech nerve

$$\left(\cdots \overrightarrow{\overrightarrow{\overrightarrow{\to }}}Y{\times }_{X}Y{\times }_{X}Y\overrightarrow{\overrightarrow{\to }}Y{\times }_{X}Y\overrightarrow{\to }Y\stackrel{\pi }{\to }X\right).$$\left( \cdots \stackrel{\to}{\stackrel{\to}{\stackrel{\to}{\to}}} Y \times_X Y \times_X Y \stackrel{\to}{\stackrel{\to}{\to}} Y \times_X Y \stackrel{\to}{\to} Y \stackrel{\pi}{\to} X \right) \,.

Taking function algebras of this yields, by the above,

$$\left(\cdots \overleftarrow{\overleftarrow{\overleftarrow{\leftarrow }}}S{\otimes }_{R}S{\otimes }_{R}S\overleftarrow{\overleftarrow{\leftarrow }}S{\otimes }_{R}S\overleftarrow{\leftarrow }S\stackrel{{\pi }^{*}}{\leftarrow }R\right).$$\left( \cdots \stackrel{\leftarrow}{\stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}}} S \otimes_R S \otimes_R S \stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}} S \otimes_R S \stackrel{\leftarrow}{\leftarrow} S \stackrel{\pi^*}{\leftarrow} R \right) \,.

Writing again $C=S{\otimes }_{R}S$ for the Sweedler coring, this is

$$\left(\cdots \overleftarrow{\overleftarrow{\overleftarrow{\leftarrow }}}C{\otimes }_{S}C\overleftarrow{\overleftarrow{\leftarrow }}C\overleftarrow{\leftarrow }S\stackrel{{\pi }^{*}}{\leftarrow }R\right).$$\left( \cdots \stackrel{\leftarrow}{\stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}}} C \otimes_S C \stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}} C \stackrel{\leftarrow}{\leftarrow} S \stackrel{\pi^*}{\leftarrow} R \right) \,.

Descent in terms of coring comodules

Given a morphism $f:R\to S$ with corresponding Sweedler coring $(C=S{\otimes }_{R}S,\Delta ,ϵ)$ as above, the category of descent data $\mathrm{Desc}(S/R)$ for the categories of right modules along $k$-algebra extension $R\to S$ is precisely the category of right $C$-comodules.

In other words, the objects of $\mathrm{Desc}(S/R)$ are the pairs $(N,\alpha )$ where $N$ is a right $S$-module, and $\alpha :N\to N{\otimes }_{R}S$ is a right $S$-module morphism and if we write $\alpha (m)={\sum }_i{m}_i\otimes s_i$ then

• ${\sum }_i\alpha (m_i)\otimes s_i={\sum }_i{m}_i\otimes 1\otimes s_i$,

• ${\sum }_i{m}_i{s}_i=m$.

In terms of (co)monadic descent

This coring-formulation of descent may be understood as special case of comonadic descent (see also the discussion at Bénabou–Roubaud theorem). We spell this out in detail:

The bifibration in question is

$p:$ Mod $\to$ CRing

that sends an object in the category Mod of modules to the ring that it is a module over.

A descent datum for a morphism $f:R\to S$ with respect to this bifibration is a (co)algebra object over the comonad $f_{*}{f}^{*}$ induced by this morphism. We have that

• the morphism $f_{*}$ sends an $R$-module $N$ to the $S$-module $N{\otimes }_{R}S$;

• the morphism $f^{*}$ sends an $S$-module $P$ to the $R$-module $P{\otimes }_S{S}_R$, where $S_R$ is $S$ regarded as a left $S$- and a right $R$-module. So $P{\otimes }_S{S}_R$ is just the $S$-module $P$ with only the right $R$-action remembered.

Accordingly, the comonad with underlying functor $f^{*}{f}_{*}$ sends an $S$-module $P$ to the $S$-module $P{\otimes }_{S}S{\otimes }_{R}S=P{\otimes }_{S}C$.

A (co)algebra object for this comad is hence a co-action morphism

$$P\to P{\otimes }_{S}C$$P \to P \otimes_S C

compatible with the monad action. This is precisely a comodule over the Sweedler coring, as defined above.

Geometric interpretation

Descent for Sweedler corings is a special case of comonadic descent. We describe this in detail and relate it by duality to the geometrically more intuitive monadic descent for codomain fibrations.

Assuming again a suitable geometric context as above, we may identify a module over $R=C(X)$ with (the collection of sections of) a vector bundle (or rather a suitable generalization of that: a coherent sheaf) over $X$. Similarly for $Y$. So we write

$$\mathrm{Vec}(X):=R\mathrm{Mod}$$Vec(X) := R Mod

and

$$\mathrm{Vect}(Y):=S\mathrm{Mod}$$Vect(Y) := S Mod

for the corresponding categories of modules. The assignment of such categories to spaces

$$\mathrm{Vec}:Z\mapsto \mathrm{Vec}(Z)$$Vec : Z \mapsto Vec(Z)

extends to a contravariant pseudofunctor

$$\mathrm{Vec}:{\mathrm{Spaces}}^{\mathrm{op}}\to \mathrm{Cat}$$Vec : Spaces^{op} \to Cat

by assigning to a morphism $f:Y\to X$ of spaces the corresponding functor

$$\mathrm{Vec}(X)\simeq C(X)\mathrm{Mod}\stackrel{-{\otimes }_{f}C(Y)}{\to }C(Y)\mathrm{Mod}\simeq \mathrm{Vec}(Y).$$Vec(X) \simeq C(X) Mod \stackrel{- \otimes_{f} C(Y)}{\to} C(Y) Mod \simeq Vec(Y) \,.

This way $\mathrm{Vec}$ becomes a prestack of categories on our category of spaces.

If this prestack satisfies descent along suitable covers, it is a stack.

Geometrically this is the case if for each morphism $\pi :Y\to X$ that is regarded as a cover, the category $\mathrm{Desc}(Y,\mathrm{Vec})$ whose objects are tuples consisting of

• an object $a\in \mathrm{Vec}(Y)$

• an isomorphism $g:{\pi }_1^{*}a\to {\pi }_2^{*}a$

• such that

  $$\begin{array}{ccc}& & {\pi }_2^{*}a\\ & {}^{{\pi }_{12}^{*}g}\nearrow & & {\searrow }^{{\pi }_{23}^{*}g}\\ {\pi }_1^{*}a& & \stackrel{{\pi }_{13}^{*}}{\to }& & {\pi }_3^{*}\end{array}$$\array{ && \pi_2^* a \\ & {}^{\mathllap{\pi_{12}^* g}}\nearrow && \searrow^{\mathrlap{\pi_{23}^* g}} \\ \pi_1^* a &&\stackrel{\pi_{13}^*}{\to}&& \pi_3^* }

  commutes.

Morphism are defined similarly (see stack and descent for details).

To get the geometric pucture that underlies, by duality, the above comodule definition of descent, we need to reformulate this just a little bit more:

every ordinary vector bundle $E\to X$ (of finite rank) is the associated bundle $E\simeq P{\times }_{O(n)}V$ of an O(n)-principal bundle $P\to X$, and as such its sections may be identified with $O(n)$-equivariant functions $P\to V\simeq {ℝ}^n$ on the total space of $P$.

Using this we may think of the $C(X)$-module of sections of $E$ as a submodule of the $C(X)$-module of all functions on $P$

$$\Gamma (E)\subset C(P).$$\Gamma(E) \subset C(P) \,.

We now reformulate the geometric descent for vector bundles in terms of geometric descent for their underlying principal bundles, and then take functions on everything in sight to obtain the comodule definition of descent that we want to describe:

A descent datum (transition function) for a principal bundle $Q\to Y$ may be thought of as the the morphism $g$ in the double pullback diagram

$$\begin{array}{ccccc}& & Y{\times }_{X}Y{\times }_{Y}Q& \to & Q\\ & & \downarrow & & \downarrow \\ & {}^g\swarrow & Y{\times }_{X}Y& \to & Y\\ & & \downarrow & & {\downarrow }^{\pi }\\ Q& \to & Y& \stackrel{\pi }{\to }& X\end{array}.$$\array{ &&Y\times_X Y \times_Y Q &\to& Q \\ &&\downarrow && \downarrow \\ &{}^{\mathllap{g}}\swarrow&Y \times_X Y &\to& Y \\ &&\downarrow && \downarrow^{\mathrlap{\pi}} \\ Q &\to& Y &\stackrel{\pi}{\to}& X } \,.

Because here $Y{\times }_{X}Y{\times }_{Y}Q$ is the space whose points consist of a point in a double overlap of the cover and a point in the fiber of $Q$ over that with respect to one patch, and the morphism identifies this with a point in the fiber of $Q$ regarded as sitting over the other patch. Analogously, there is a cocycle condition on $g$ on triple overlaps.

Now, blindly applying our functor that takes functions of spaces to the above diagram yields the double pushout diagram

$$\begin{array}{ccccc}& & C(Y{\times }_{X}Y{\times }_{Y}Q)& \leftarrow & C(Q)\\ & & \uparrow & & \uparrow \\ & {}^{g^{*}}\nearrow & C(Y{\times }_{X}Y)& \leftarrow & C(Y)\\ & & \uparrow & & {\uparrow }^{{\pi }^{*}}\\ C(Q)& \leftarrow & C(Y)& \stackrel{{\pi }^{*}}{\leftarrow }& C(X)\end{array}.$$\array{ &&C(Y\times_X Y \times_Y Q) &\leftarrow& C(Q) \\ &&\uparrow && \uparrow \\ &{}^{\mathllap{g}^*}\nearrow&C(Y \times_X Y) &\leftarrow& C(Y) \\ &&\uparrow && \uparrow^{\mathrlap{\pi^*}} \\ C(Q) &\leftarrow& C(Y) &\stackrel{\pi^*}{\leftarrow}& C(X) } \,.

We may restrict to $N:=\Gamma (E)\subset C(Q)$ as just discussed and switch to the notation from above to get

$$\begin{array}{ccccc}& & N{\otimes }_{S}C& \leftarrow & N\\ & & \uparrow & & \uparrow \\ & {}^{g^{*}}\nearrow & C& \leftarrow & S\\ & & \uparrow & & {\uparrow }^{{\pi }^{*}}\\ N& \leftarrow & S& \stackrel{{\pi }^{*}}{\leftarrow }& R\end{array}.$$\array{ &&N \otimes_{S} C &\leftarrow& N \\ &&\uparrow && \uparrow \\ &{}^{\mathllap{g}^*}\nearrow& C &\leftarrow& S \\ &&\uparrow && \uparrow^{\mathrlap{\pi^*}} \\ N &\leftarrow& S &\stackrel{\pi^*}{\leftarrow}& R } \,.

The morphism

$$\alpha :=g^{*}:N\to N{\otimes }_{S}C$$\alpha := g^* : N \to N \otimes_S C

obtained this way is the co-action morphism from the above algebraic definition.

The further cocycle condition on $g$ similarly translates into the condition that $\alpha$ really satisfies the comodule property.

Properties are related to the ones of the ring extension

Various properties of canonical coring correspond to adequate properties of the ring extension. For example, coseparable Sweedler corings correspond to split extension?s (the $k$-algebra extension $R\to S$ is split if there is an $R$-bimodule map $h:S\to R$ with $h(1_S)=1_R$).

Related concepts

• descent

  • cover

  • cohomological descent

  • monadic descent,

    • Sweedler coring

    • higher monadic descent

    • descent in noncommutative algebraic geometry

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

References

Sweedler corings are named after Moss Sweedler.