From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4854 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Enrichment over a monoidal bicategory Date: Wed, 20 May 2009 15:12:08 +0100 (BST) Message-ID: Reply-To: Richard Garner NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1242925310 14446 80.91.229.12 (21 May 2009 17:01:50 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 21 May 2009 17:01:50 +0000 (UTC) To: Alex Hoffnung , categories@mta.ca Original-X-From: categories@mta.ca Thu May 21 19:01:43 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1M7BeB-0000CH-5X for gsmc-categories@m.gmane.org; Thu, 21 May 2009 19:01:35 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1M7As2-0003Yg-Hz for categories-list@mta.ca; Thu, 21 May 2009 13:11:50 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4854 Archived-At: Dear Alex, A fair amount of the theory of enriched bicategories is worked out in Steve Lack's PhD thesis "The algebra of distributive and extensive categories". I don't think there have been any further attempts to develop the theory to any serious degree. Best wishes, Richard --On 19 May 2009 22:10 Alex Hoffnung wrote: > Hi > > I have found that there is a fairly straightforward way to generalize > the notion of enrichment over a monoidal category to enrichment over a > monoidal bicategory. Namely, a "bicategory enriched over a monoidal > bicategory V" consists of the following: > > 1) a collection of "objects" A, B, C,... > > 2) for any pair of objects A,B, an object in V called hom(A,B) > > 3) for any triple of objects A,B,C a morphism in V called composition: > hom(A,B) tensor hom(B,C) -> hom(A,C) > where "tensor" is the tensor product in V. > > 4) for any object A a morphism in V called identity: I_A -> hom(A,A) > > 5) for any quadruple of objects A,B,C,D a 2-isomorphism in V called > the associator, which does the obvious thing. > > plus left and right unitors, and so on with all the axioms closely > following those of the definition of a bicategory. > > I am looking to be pointed in the right direction in the literature. > Can anyone help? I am aware of the fc-multicategories by Leinster and > earlier work by Walters, but those do not seem to use the monoidal > structure to enrich as I want. > > Best, > Alex Hoffnung > > >