From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4849 Path: news.gmane.org!not-for-mail From: Alex Hoffnung Newsgroups: gmane.science.mathematics.categories Subject: Enrichment over a monoidal bicategory Date: Tue, 19 May 2009 22:10:51 -0500 Message-ID: Reply-To: Alex Hoffnung NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1242830839 10014 80.91.229.12 (20 May 2009 14:47:19 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 20 May 2009 14:47:19 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Wed May 20 16:47:12 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 1M6n4X-0006PY-Gn for gsmc-categories@m.gmane.org; Wed, 20 May 2009 16:47:09 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1M6mBv-0001Tr-GR for categories-list@mta.ca; Wed, 20 May 2009 10:50:43 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4849 Archived-At: 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