Date: Tue, 16 Jan 1996 14:06:43 -0400 (AST) Subject: bimodules in a biclosed category Date: Tue, 16 Jan 1996 08:11:29 -0500 From: Michael Barr Has anyone proved that if you take an "algebra" (actually monoid) object in a monoidal biclosed category that has equalizers and coequalizers, then the category of two-sided modules for that algebra is again a monoidal biclosed category. Mac Lane did this in 1965 when everything is symmetric (the tensor, the algebra and the modules) under the (surely irrelevant) assumption that the original category is also abelian. The fact is certainly true, but writing down the proof would be rather painful. Michael Barr Date: Wed, 17 Jan 1996 10:13:43 -0400 (AST) Subject: Re: bimodules in a biclosed category Date: Wed, 17 Jan 1996 10:42:32 +0100 (MET) From: koslowj@iti.cs.tu-bs.de Hello, Regarding Michael Barr's question: I have this result for bi-closed bicategories, and actually talked about it in Halifax last year. Als Micheal rightly suspects, some of the proofs get tedious, and designing the relevant diagrams is something else, even using xy-pic! I hope that the paper will be finished by March. Best regards, -- J"urgen Koslowski Institut f"ur theoretische Informatik TU Braunschweig Braunschweig, Germany Date: Wed, 17 Jan 1996 10:14:22 -0400 (AST) ubject: Re: bimodules in a biclosed category Date: Wed, 17 Jan 1996 12:22:32 +0000 From: Steve Vickers >Has anyone proved that if you take an "algebra" (actually monoid) object >in a monoidal biclosed category that has equalizers and coequalizers, >then the category of two-sided modules for that algebra is again a >monoidal biclosed category. Mac Lane did this in 1965 when everything >is symmetric (the tensor, the algebra and the modules) under the (surely >irrelevant) assumption that the original category is also abelian. The >fact is certainly true, but writing down the proof would be rather >painful. The question made me think - as perhaps it was meant to - of modules over non-commutative rings, and I wondered whether its scope was unnecessarily restricted in considering just 2-sided modules over a single algebra. We also know that there can be a rich and well-behaved structure for bimodules over two algebras: (A,B)-bimod tensor (B,C)-bimod is (A,C)-bimod B (A,B)-bimod hom (A,C)-bimod is (B,C)-bimod A (A,B)-bimod hom (C,B)-bimod is (C,A)-bimod B Does anyone know how to state the question to cover this more general context? But then what works for algebras ought also to work for algebroids, i.e. enriched categories. Steve Vickers. Date: Wed, 17 Jan 1996 11:14:36 -0400 (AST) Subject: Re: bimodules in a biclosed category Date: Wed, 17 Jan 1996 10:00:03 -0500 From: Michael Barr - - Date: Wed, 17 Jan 1996 12:22:32 +0000 - From: Steve Vickers - - >Has anyone proved that if you take an "algebra" (actually monoid) object - >in a monoidal biclosed category that has equalizers and coequalizers, - >then the category of two-sided modules for that algebra is again a - >monoidal biclosed category. Mac Lane did this in 1965 when everything - >is symmetric (the tensor, the algebra and the modules) under the (surely - >irrelevant) assumption that the original category is also abelian. The - >fact is certainly true, but writing down the proof would be rather - >painful. - - The question made me think - as perhaps it was meant to - of modules over - non-commutative rings, and I wondered whether its scope was unnecessarily - restricted in considering just 2-sided modules over a single algebra. We - also know that there can be a rich and well-behaved structure for bimodules - over two algebras: - - (A,B)-bimod tensor (B,C)-bimod is (A,C)-bimod - B - - (A,B)-bimod hom (A,C)-bimod is (B,C)-bimod - A - - (A,B)-bimod hom (C,B)-bimod is (C,A)-bimod - B - - Does anyone know how to state the question to cover this more general - context? But then what works for algebras ought also to work for - algebroids, i.e. enriched categories. - - Steve Vickers. - - - - Actually, if the ground tensor is not symmetric, you cannot even define the notion of (A,B) bimodule. You can define that of left A, right B bimodule and that seems to be that. Perhaps that is what Steve means when he says (A,B) bimodule. In that case, he is, of course, correct. In fact, in writing this up, I find it less confusing to do the more general situation exactly as described. But I would rather not have to write it up if someone has already done it as it is rather unpleasant. Michael Date: Thu, 18 Jan 1996 12:03:20 -0400 (AST) Subject: bimodules in a biclosed category Date: Thu, 18 Jan 1996 10:38:59 +1100 From: Ross Street In response to Michael Barr's question: Theorem: If V is a closed braided monoidal category which is complete and cocomplete then the bicategory V-Mod of V-categories, V-modules (sometimes called V-bimodules, V-distributors or V-profunctors), and V-module morphisms is a monoidal bicategory (meaning the hom of a tricategory with one object). However, in order for V-Mod to be braided, V must be symmetric in which case V-Mod is also symmetric (also called strongly involutory by Baez-Dolan). If you are only interested in monoids in V, just take the subbicategory of one-object V-categories. Of course, then, you can do with less completeness and cocompleteness on V. But yes, the detailed proof of this makes a long, but fairly routine, story. Brian Day and I are working on a paper "Monoidal bicategories and Hopf algebroids" which will contain a discreet amount of detail (along with other things). I have been talking about aspects of the paper in our Category Seminars. Regards, Ross Date: Wed, 24 Jan 1996 20:26:55 -0400 (AST) Subject: More on Mike Barr's question Date: Thu, 25 Jan 1996 10:00:04 +1100 From: Ross Street There has been some further discussion between Mike Barr and me on modules between monoids in a complete, cocomplete, closed monoidal category. Mike responded: >If I have understood your theorem, it is claimed only for the case that >the original category is braided. I was interested in the genuinely >asymmetric case, so this doesn't apply. Although your arguments >are probably valid in that case, at a guess. Have I missed something? So I said: It looks as though **I** missed something. I didn't include the older fact well known to enriched category theorists: even without any symmetry or braiding, V-Mod is a bicategory in which all right extensions and right liftings exist (I don't like the word "biclosed"; I use "left closed", "right closed" and "closed" for both when dealing with a monoidal category). In particular, for any V-category A, each hom category V-Mod(A,A) is a closed monoidal category under composition (= tensor product) of modules (as before, I don't say "bimodules" either). Even more particularly you can take A to be a monoid in V. For this there ARE references. I believe Benabou, in his Louvain-la-neuve notes "Les Distributeurs" Rapport No 33 jan 1973, only looked at the case of V symmetric. Same is true of Bill Lawvere's "Metric spaces" paper. But it is not hard to generalise. My paper Enriched categories and cohomology, Quaestiones Math. 6 (1983) 265-283; MR85e:18007 does the non-symmetric case. But it does it more generally for V a bicategory (without commutativity you might as well have "several objects"). Other papers which build on this are: Cauchy characterization of enriched categories, Rendiconti del Seminario Matematico e Fisico di Milano 51 (1981) 217-233; MR85e:18006. (with R. Betti, A. Carboni, and R. Walters) Variation through enrichment, J. Pure Appl. Algebra 29 (1983) 109-127; MR85e:18005. (with A. Carboni, S. Johnson and D. Verity) Modulated bicategories, J. Pure Appl. Algebra 94 (1994) 229-282. Gordon and Power have also done Gabriel-Ulmer duality for W-Mod where W is a decent bicategory (homs locally presentable). JPAA? The recent thing with Brian Day looks at when V-Mod itself is "monoidal", a natural step beyond the above references & studied by Carboni, Walters et al in the case where V is a poset (or W is locally a poset). Best wishes, Ross