17
$\begingroup$

I have two closely related questions:

  1. Who first recognized that Morita equivalence was equivalence in the bicategory of Rings, Bimodules, and Intertwiners?

  2. I've heard this bicategory called the "Morita bicategory". Who first called it this?


Here's what I've been able to figure out:

Morita introduced what now is called "Morita equivalence" in

Kiiti Morita. Duality for modules and its applications to the theory of rings with minimum condition. Sci. Rep. Tokyo Kyoiku Daigaku Sect. A, 6:83–142, 1958. MR0096700

He defined it in terms of equivalences of categories of modules, but proved that all equivalences of categories of modules are implemented by tensor product with a bimodule. (Also by homming from a bimodule.) It's worth also noting that Morita primarily discussed contravariant equivalences between various categories of modules satisfying some size conditions, rather than covariant equivalences between categories of all modules; his primary interest was on "dualities" $\hom_A(-,U)$ and $\hom_B(-,U)$ where $U$ is a fixed $A$-$B$-bimodule.

Eilenberg and Watts independently proved the theorem now named after them jointly, that colimit-preserving functors between categories of modules are given by tensoring with bimodules:

Charles E. Watts, Intrinsic characterizations of some additive functors, Proc. Amer. Math. Soc. 11, 1960, 5–8, MR0806.0832

Samuel Eilenberg, Abstract description of some basic functors, J. Indian Math. Soc. (N.S.) 24, 1960, 231–234, MR0125148

Bass gave an elegant survey of Morita's work and the Eilenberg–Watts theorem, focusing on applications to Wedderburn structure theory and the Brauer group in:

Hyman Bass, The Morita Theorems, Lectures given at the University of Oregon, 1962. PDF

That paper uses (and, as far as I can tell, introduces) the phrase "Morita context" for a pair of bimodules $_A P_B$ and $_B Q_A$ and bimodule homomorphisms $P \otimes_B Q \to A$ and $Q \otimes_A P \to B$, not necessarily invertible but satisfying that the two natural maps $P \otimes_B Q \otimes_A P \rightrightarrows P$ agree, as do the two natural maps $Q \otimes_A P \otimes_B Q \rightrightarrows Q$. I feel like I don't see the phrase "Morita context" very much any more, but perhaps this is due to a biased sample on my part.

The terms "Morita equivalence" and "Morita-invariant" appear (perhaps for the first time?) in

P.M. Cohn, Morita equivalence and duality, Queen Mary College Mathematics Notes, Queen Mary College, London, 1968 (lectures written 1966, republished with additional citations 1976). MR0258885

The notion of bicategory was introduced in:

J. Bénabou. Introduction to bicategories, part I. Reports of the Midwest Category Seminar, Lecture Notes in Mathematics 47, pages 1-77. Springer, 1967. MR0220789

He discusses in some detail the bicategory of Rings, Bimodules, and Intertwiners, and observes that it receives a functor (named "modulation" by Tang, Weinstein, and Zhu, 2007) from the category of Rings and Homomorphisms.

The canonical reference for Morita theory is another work by Bass:

Hyman Bass, Algebraic K-Theory, Benjamin, 1968. MR0249491

Notably, "Chapter II: Categories of Modules and their Equivalences". This book never uses the language of bicategories, which should not surprise: the book came out in 1968, which means it went to the publisher by 1967, which means that Bass at the time of writing didn't have Bénabou's paper. The book does, however, make the following observation: There is a (strict 1-) category whose objects are Rings and whose morphisms are Isomorphism Classes Of Bimodules, and Morita equivalence (not a term used in the book) is isomoprhism in that category. This almost answers Question 1, but not quite.

I then lose track of the citations, which seem to multiply quickly. It's clear that all ingredients were available by the end of the 1960s and reasonably well known by the mid 1970s (e.g. Bunge, 1979 was doing quite sophisticated category theory). Bounding from the other end, by Brouwer, 2003 the observation that Morita equivalence is best understood in terms of bicategories was routine.

$\endgroup$
  • $\begingroup$ Duskin may have been the first to say it explicitly, but I can't supply a reference at the moment. John Baez's twf has a bit on this and may have one. Otherwise there's amore recent paper of Ross Street that may reference Duskin $\endgroup$ – David Roberts Dec 10 '15 at 6:18
  • 2
    $\begingroup$ See Street's response at the bottom of math.ucr.edu/home/baez/week209.html, for instance $\endgroup$ – David Roberts Dec 10 '15 at 6:28
  • $\begingroup$ The link to Bass's notes has evaporated, do you still have them? $\endgroup$ – David Roberts Apr 12 at 10:13
  • 1
    $\begingroup$ @DavidRoberts This is the problem with changing institutions. Try here. My plan is to maintain that domain going forward, although I can't promise that I won't think it a good idea to do a massive reorganization five years from now. $\endgroup$ – Theo Johnson-Freyd Apr 12 at 15:43
  • $\begingroup$ thanks for that! $\endgroup$ – David Roberts Apr 13 at 0:21
3
$\begingroup$

Let me venture an answer to (2), a rather recent source, but it's the oldest I have found:

  1. The name "Morita bicategory" was introduced in January 2008 by Hellen Colman, in a talk at the Max Kelly Conference in Category Theory (Cape Town, 21–26 January, 2008): Lusternik-Schnirelmann theory for the Morita bicategory of Lie groupoids.
    She subsequently abandonded the name, see this arXiv paper.
$\endgroup$
  • $\begingroup$ Other names use 'Brauer', as the equivalence classes of objects are the Brauer group of the ring. $\endgroup$ – David Roberts Dec 10 '15 at 11:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.