From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2414 Path: news.gmane.org!not-for-mail From: Ross Street Newsgroups: gmane.science.mathematics.categories Subject: Re: monoidal terminology Date: Tue, 19 Aug 2003 10:52:42 +1000 Message-ID: References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" ; format="flowed" X-Trace: ger.gmane.org 1241018642 4033 80.91.229.2 (29 Apr 2009 15:24:02 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:24:02 +0000 (UTC) To: categories@mta.ca, jean benabou Original-X-From: rrosebru@mta.ca Tue Aug 19 19:44:55 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 19 Aug 2003 19:44:55 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19pF94-0007bC-00 for categories-list@mta.ca; Tue, 19 Aug 2003 19:40:06 -0300 X-Sender: street@icsmail.ics.mq.edu.au In-Reply-To: Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 6 Original-Lines: 47 Xref: news.gmane.org gmane.science.mathematics.categories:2414 Archived-At: Dear Jean >Suppose V is a (fixed) monoidal symmetric category and C is a category >enriched over V . > >The following notions should be "obviously well-known" , but I cannot find >any reference for them in the "standard literature" , do such references >exist ? > >1- A monoidal structure on C (of course, as an enriched category) >2- A symmetric monoidal structure on C >3- A closed monoidal structure on C As you expected these concepts are truly well known. I can give two references that I would consider part of the "standard literature", at the two ends of a chronological spectrum: [1] B.J. Day, On closed categories of functors, Lecture Notes in Math 137 (Springer, 1970) 1-38 [2] B.J. Day, P. McCrudden and R. Street, Dualizations and antipodes, Applied Categorical Structures 11 (2003) 229-260 In [1] you will find the notion of a "premonoidal V-category" A which, because of your term "profunctor", was renamed "promonoidal V-category". This paper contains part of Brian Day's PhD thesis. It carefully explains explicitly how a monoidal V-category is promonoidal, and what it means for it to be closed. It also carefully defines symmetry for promonoidal V-categories (and shows how it amounts to a symmetry for the convolution monoidal structure on the V-category [A,V] of V-functors A --> V). In [2] you will find monoidal (or pseudomonoid) structures, together with closed, symmetric and braided ones, on objects in any autonomous monoidal bicategory (such as V-Mod, V-Prof or V-Dist, whichever name you prefer). For the three matters in question, this is perhaps an improvement on B.J. Day and R. Street, Monoidal bicategories and Hopf algebroids, Advances in Math. 129 (1997) 99-15. Best regards, Ross