The n-Category Café

I think Heyneman was in on the early development
jointly or independently?? of what is now known only as Sweedler notation

Though not much interested in the name-related terminology, I could say that the descent for (the fibered category of modules over) noncommutative algebras and the conceptTh of corings/cocategories were certainly known as examples of general machineries in category theory in 1960s, but specific study of Sweedler coring and of corings got specific attention and tangible publications were probably starting with him.

The descent for noncommutative algebras is a special case of comonadic descent which was studied much in 1960s (Beck etc.) and it was known at the time that one of the basic examples is the extension of algebras/rings and that commutativity is not critical. Still a very much cited reference is by Cippola from 1970s which describes the descent for noncommutative rings in elementary terms (without corings and comonads) parallel to the treatment in SGA I. The comonad for descent of noncommutative algebras is the tensoring with Sweedler coring, of course. The descent for modules over noncommutative algebras for covers by Gabriel localizations was considered in 1988 book of A. L. Rosenberg in Leites’ Stockholm seminars volume and then again independently by van Oystaeyen and his school in early 1990s and in both references unfortunately systematics viewpoint of comonadic descent has been obscured by elementary proofs using specifics of localization theory. 1988 Rosenberg’s Compositio paper *Noncommutative schemes* systematically takes the point of view that the relative generalization of category of quscicoherent sheaves over a general scheme (with relative functor to quasicoherent sheaves over the base scheme) should be axiomatized and studied with emphasis on exactness, as well as monadicity and comonadicity properties with the central role played by applications of Beck’s monadicity theorem. This fundamental application of monadicity and category theory to noncommutative algebraic geometry has not been much noticed by the Hopf algebraic community which needed the more basic understanding of Hopf-Galois descent and coalgebra Galois descent and generalizations. In Spring 2002 I have talked to Valery Lunts who proposed to me that the analogue of modules over Borel construction for the comodule algebras should replace the definition of relative Hopf modules, making the descent theorems like Schneider’s obvious (the basic ideas of this are explained in much later paper of mine “Some equivariant constructions in noncommutative geometry”). Also in 2002, Pjotr Hajac proposed that Sweedler’s corings should be used for the same purpose – in fact the modules over “coborel constuction” and modules over the associated coring are just two viewpoints on the same thing. Then the whole subject of corings in Hopf context got developed much further by Tomasz Brzeziński, Gabi Böhm etc. and influenced by the study of mixed distributive laws which was much studied in this context few years before. Cf. also [descent in noncommutative algebraic geometry](http://ncatlab.org/nlab/show/descent+in+noncommutative+algebraic+geometry).