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This is intended to continue the issues discussed in the Lafforgue thread!
I have added an idea section to Morita equivalence where I sketch what I perceive to be the overarching pattern stressing in particular the two completion processes involved. I worked with ’hyphens’ there but judging from a look in Street’s quantum group book the pattern can be spelled out exactly at a bicategorical level.
I might occasionally add further material on the Morita theory for algebraic theories where especially the book by Adamek-Rosicky-Vitale (pdf-draft) contains a general 2-categorical theorem for algebraic theories.
Another thing that always intrigued me is the connection with shape theory where there is a result from Betti that the endomorphism module involved in ring Morita theory occurs as the shape category of a ring morphism in the sense of Bourn-Cordier. Another thing worth mentioning on the page is that the Cauchy completion of a ring in the enriched sense is actually its cat of modules (this is in Borceux-Dejean) - this brings out the parallel between Morita for cats and rings.
As one of the founders of shape theory (way back!!!!!), let me point out that some of the results were proved by someone called Morita! (There may be more than one such.)
I added few links (especially Bass’s book chapter 2 which is so brightly written and only 2-3 old fashioned names of the terms separate it from being entirely modern despite 45 years since it was published) and I also mentioned Morita context. Here also archiving an old query from Morita equivalence:
Dmitri Pavlov: Tsit-Yuen Lam in his book “Lectures on modules and rings” on pages 488 and 489 states the Morita equivalence theorem using progenerators (i.e., finitely generated projective generators) instead of just generators. Are these two versions equivalent?
Dmitri Pavlov: I would like to state the Morita equivalence theorem as a 2-equivalence between two bicategories: The bicategory of rings, bimodules and their intertwiners and the bicategory of abelian categories that are equivalent to the category of modules over some ring (i.e., abelian categories that have all small coproducts and a compact projective generator), Eilenberg-Watts functors between these categories (i.e., right exact functors that commute with direct sums) and natural transformations. Is it possible to do this and what is the precise statement then?
My apologies if I don’t give credit in the appropriate way! I am aware that for a change a lot of Morita theory actually is due to Morita and your remark reminded me that some shape theory is so too but judging from your book’s bibliography that seems to concern rather shape of topological spaces than module theory.
The bicategorical brush up that Betti gave to shape theory in the 1984 cahiers paper nevertheless was it that brought the connection between shape theory and module theory to my awareness. I’ve actually brought this up here in the hope that a first rate expert like you might clarify this connection with a remark in the entry on Morita equivalence.
I generally have the impression that a lot of what I perceive somewhatly hazily as an ’overarching pattern’ concerning Morita theory is actually well understood by experts and that the community would greatly profit from making these insights more widely available.
Unfortunately I forget what was in Renato Betti’s paper.
@zskoda: very helpful additions indeed, thanks!
@Tim_Porter: The section of Betti concerning applications of shape theory to module theory is sec. 3. He works in an Abelian group enriched setting and considers a morphism of (unital) rings $K:A \rightarrow T$. the claim is then that the shape category $S_K$ is the endomorphism ring $End_A T$ of T considered as a left A-module. He then states a theorem that is apparently a variant of a result of Frei-Kleisli that if T as A-module is finitely generated projective every shape invariant shape T-module is a right kan extension along K. I guess my idea was that this control of shape invariance as kan extension corresponds precisely to ’convergence’ of a morita context for A and T to a morita equivalence. At the moment I don’t see how exactly $End_A T$ fits into Morita though so nevermind if you don’t want to invest your time in a vague suggestion. In any way it might be necessary to take into account the newer papers at Morita context to figure out how the shape category fits into this if it does!
made the requested Cauchy completion redirect to Cauchy complete category
creating equivalence in a 2-category made me look again at the entry Morita equivalence. I have now expanded the Idea-section there, adding a lead-in paragraph that first says what classical Morita equivalence actually is, before entering the discussion of its vast generalizations.
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