Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory kan lie-theory limit limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology newpage nlab nonassociative noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorThomas Holder
    • CommentTimeAug 3rd 2014
    • (edited Aug 3rd 2014)

    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.

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeAug 3rd 2014

    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.)

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeAug 3rd 2014
    • (edited Aug 3rd 2014)

    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?

    • CommentRowNumber4.
    • CommentAuthorThomas Holder
    • CommentTimeAug 3rd 2014

    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.

    • CommentRowNumber5.
    • CommentAuthorTim_Porter
    • CommentTimeAug 4th 2014

    Unfortunately I forget what was in Renato Betti’s paper.

    • CommentRowNumber6.
    • CommentAuthorThomas Holder
    • CommentTimeAug 4th 2014
    • (edited Aug 4th 2014)

    @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:ATK:A \rightarrow T. the claim is then that the shape category S KS_K is the endomorphism ring End ATEnd_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 AT 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!

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeAug 10th 2014

    made the requested Cauchy completion redirect to Cauchy complete category

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 4th 2017
    • (edited May 4th 2017)

    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.

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeNov 22nd 2019

    I added several references on the Hopf algebra case.

    diff, v35, current

  1. Add the definition of Morita equivalence between fusion categories.

    Z. A. Jia

    diff, v38, current

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 30th 2020

    reversed tensor produce x revy:=xyx\otimes^{rev}y:=x\otimes y.

    Did you not mean yxy\otimes x?

    diff, v40, current

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 30th 2020

    Obviously the answer is ’Yes’, so I’ve changed it.

    diff, v41, current

    • CommentRowNumber13.
    • CommentAuthorTim_Porter
    • CommentTimeNov 7th 2021

    Put links from authors names.

    diff, v43, current

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)