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A canonical or 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.
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
with coproduct
given by the bilinear extension of the formula
and counit
The element $1\otimes 1$ is a grouplike element in the Sweedler’s coring.
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
is the tensor product of the functions on the factors:
Then let $\pi : Y \to X$ be a morphism of spaces and set
and
and
The morphism $\pi$ induces its augmented Čech nerve
Taking function algebras of this yields, by the above,
Writing again $C = S \otimes_R S$ for the Sweedler coring, this is
Various properties of canonical coring correspond to adequate properties of the ring extension. For example, coseparable canonical 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 $h(1_S) = 1_R$).
In the case of the trivial ring extension $R\to R$ the coproduct of the canonical coring is the canonical isomorphism $R\to R\otimes_R R$ and the counit is the identity $R\to R$. Thus, every right $R$-module is a comodule over the canonical coring. Such a canonical coring is called a trivial coring. This example puts the rings (or associative unital algebras) and their categories of modules as a special case of corings and their categories of comodules. In view of the next paragraph, this is a generalization of a coring of a cover for the case of a trivial (identity) cover.
Given a morphism $f : R \to S$ with corresponding Sweedler coring $(C = S \otimes_R S,\Delta,\epsilon)$ 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
$\sum_i \alpha(m_i)\otimes s_i = \sum_i m_i\otimes 1\otimes s_i$,
$\sum_i m_i s_i = m$.
This coring-formulation of descent may be understood as special case of comonadic descent (see also the discussion at Bénabou-Roubaud theorem). See e.g. (Hess 10, section 2) for a review. We spell this out in a bit more detail:
The bifibration in question is
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 comonad is hence a co-action morphism
compatible with the monad action. This is precisely a comodule over the Sweedler coring, as defined above.
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
and
for the corresponding categories of modules. The assignment of such categories to spaces
extends to a contravariant pseudofunctor
by assigning to a morphism $f : Y \to X$ of spaces the corresponding functor
This way $Vec$ 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 $Desc(Y,Vec)$ whose objects are tuples consisting of
an object $a \in Vec(Y)$
an isomorphism $g : \pi_1^* a \to \pi_2^* a$
such that
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 $P \to V \simeq \mathbb{R}^n$ 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 $P$
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
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
We may restrict to $N := \Gamma(E) \subset C(Q)$ as just discussed and switch to the notation from above to get
The morphism
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.
Applied to E-infinity rings the Sweedler coring construction yields the Hopf algebroids of dual Steenrod algebras and appears in the Adams spectral sequence.
Sweedler corings are named after Moss Sweedler.
A textbook account is in
Section 29 there discusses the relation to the Amitsur complex and the descent theorem.
Discussion in the context of (higher) monadic descent is around example 2.24 of
Last revised on February 27, 2021 at 10:31:45. See the history of this page for a list of all contributions to it.