From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5440 Path: news.gmane.org!not-for-mail From: Marek Zawadowski Newsgroups: gmane.science.mathematics.categories Subject: a paper available Date: Tue, 29 Dec 2009 16:52:35 +0100 Message-ID: Reply-To: Marek Zawadowski NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-2; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1262182378 11942 80.91.229.12 (30 Dec 2009 14:12:58 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 30 Dec 2009 14:12:58 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Wed Dec 30 15:12:51 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 1NPzHx-0001T3-1U for gsmc-categories@m.gmane.org; Wed, 30 Dec 2009 15:12:37 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NPyrj-0001bP-GI for categories-list@mta.ca; Wed, 30 Dec 2009 09:45:31 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5440 Archived-At: The following paper Lax Monoidal Fibrations Abstract: We introduce the notion of a lax monoidal fibration and we show how it can be conveniently used to deal with various algebraic structures that play an important role in some definitions of the opetopic sets (Baez-Dolan, Hermida-Makkai-Power). We present the 'standard' such structures, the exponential fibrations of basic fibrations and three areas of applications. First area is related to the T-categories of A. Burroni. The monoids in the Burroni lax monoidal fibrations form the fibration of T-categories. The construction of the relative Burroni fibrations and free T-categories in this context, allow us to extend the definition of the set of opetopes given by T. Leinster to the category of opetopic sets (internally to any Grothendieck topos, if needed). We also show that fibration of (1-level) multicategories, considered by Hermida-Makkai-Power, is equivalent to the fibration of (finitary, cartesian) polynomial monads. This equivalence is induced by the equivalence of lax monoidal fibrations of amalgamated signatures, polynomial diagrams, and polynomial (finitary, endo) functors. Finally, we develop a similar theory for symmetric signatures, analytic diagrams (a notion introduced here), and (finitary, multivariable) analytic (endo)functors. Among other things we show that the fibrations of symmetric multicategories is equivalent to the fibration of analytic monads. We also give a characterization of such a fibration of analytic monads. An object of this fibration is a weakly cartesian monad on a slice of Set whose functor parts is a finitary functors weakly preserving wide pullbacks. A morphism of this fibration is a weakly cartesian morphism of monads whose functor part is a pullback functor. is available at http://arxiv.org/abs/0912.4464 and at my home-page http://www.mimuw.edu.pl/~zawado/papers.htm Comments are welcome. Happy New Year to all, Marek Zawadowski [For admin and other information see: http://www.mta.ca/~cat-dist/ ]