From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1145 Path: news.gmane.org!not-for-mail From: Claudio Hermida Newsgroups: gmane.science.mathematics.categories Subject: preprint: Representable Multicategories Date: Tue, 22 Jun 1999 15:53:36 +1000 Organization: School of Mathematics and Statistics, University of Sydney Message-ID: <376F24E0.2781@maths.usyd.edu.au> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241017596 29692 80.91.229.2 (29 Apr 2009 15:06:36 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:06:36 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Tue Jun 22 11:48:10 1999 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id JAA22424 for categories-list; Tue, 22 Jun 1999 09:39:04 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: Mozilla 3.01Gold (X11; I; OSF1 V4.0 alpha) Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 32 Xref: news.gmane.org gmane.science.mathematics.categories:1145 Archived-At: The preprint `Representable multicategories' is available at http://www.maths.usyd.edu.au:8000/u/hermida Abstract: We introduce the notion of representable multicategory, which stands in the same relation to that of monoidal category as fibration does to contravariant pseudofunctor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe representability in elementary terms via universal arrows. We also give a doctrinal characterisation of representability based on a fundamental monadic adjunction between the 2-category of multicategories and that of strict monoidal categories. The first main result is the coherence theorem for representable multicategories, asserting their equivalence to strict ones, which we establish via a new technique based on the above doctrinal characterisation. The other main result is a 2-equivalence between the 2-category of representable multicategories and that of monoidal categories and strong monoidal functors. This correspondence extends smoothly to one between bicategories and a localised version of representable multicategories. -- Claudio Hermida School of Mathematics and Statistics F07, University of Sydney, Sydney, NSW 2006, Australia