Special and general types Special notions Variants Extra structure Operations Theorems algebraic theory / 2-algebraic theory / (∞,1)-algebraic theory monad / (∞,1)-monad operad / (∞,1)-operad monoidal (∞,1)-category symmetric monoidal (∞,1)-category of spectra A-∞ algebra C-∞ algebra E-∞ ring, E-∞ algebra L-∞ algebra model structure on simplicial T-algebras / homotopy T-algebra model structure on operads model structure on algebras over an operad structures in a cohesive (∞,1)-topos A canonical or Sweedler coring is an algebraic structure that is (possibly a noncommutative generalization) of, 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 be the extension of associative unital -algebras (where is a commutative unital ring). The corresponding canonical coring or Sweedler coring is the -coring with coproduct given by the bilinear extension of the formula and counit The element 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 on a fiber product is the tensor product of the functions on the factors: Then let be a morphism of spaces and set and