From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5817 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: bilax_monoidal_functors?= Date: Mon, 17 May 2010 00:57:18 +0100 (BST) Message-ID: References: Reply-To: Richard Garner NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: MULTIPART/MIXED; BOUNDARY="1870870024-881670754-1274054087=:29111" Content-Transfer-Encoding: QUOTED-PRINTABLE X-Trace: dough.gmane.org 1274120107 31385 80.91.229.12 (17 May 2010 18:15:07 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 17 May 2010 18:15:07 +0000 (UTC) To: Andre Joyal Original-X-From: categories@mta.ca Mon May 17 20:15:05 2010 connect(): No such file or directory 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 4.69) (envelope-from ) id 1OE4qG-00034e-4a for gsmc-categories@m.gmane.org; Mon, 17 May 2010 20:15:04 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OE4LG-0001YA-F4 for categories-list@mta.ca; Mon, 17 May 2010 14:43:02 -0300 In-Reply-To: Content-ID: Original-Content-Type: TEXT/PLAIN; CHARSET=ISO-8859-1; format=flowed Content-ID: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5817 Archived-At: Dear Andr=E9 > In the chapter 4 of the latest version of their book > > http://www.math.tamu.edu/~maguiar/a.pdf > > Aguiar and Mahajan introduce a notion of P-monoidal functor > for P is an operad. > > ... > > The notion of P-monoidal functor for P a PROP > is not defined in their book. > > Any idea? The notion of P-monoidal functor C --> D, for P an operad,=20 may be described most expediently when D has small colimits,=20 and these distribute over tensors: for then the functor=20 category [C,D] is itself monoidal under Day convolution, and=20 a P-monoidal functor is nothing but a P-algebra in this=20 functor category. Such a definition admits an obvious generalisation to the=20 case where P is not an operad, but merely a PROP; however, it=20 seems to me that such a generalisation has no force. For=20 instance, when P is the PROP for coalgebras the notion of=20 P-monoidal functor is nothing like that of an oplax monoidal=20 functor. The problem is one of variance; in giving a=20 comultiplication F -> F * F one is required to map into a=20 coend, which is back-to-front. One way of throwing this into focus is by considering the=20 case where D has no colimits to speak of, so that the functor=20 category [C,D] is not monoidal, but merely a multicategory.=20 In any multicategory, one may still speak of a P-algebra for=20 an operad---thereby allowing the notion of P-monoidal=20 morphism of Aguiar and Mahajan to find its fully general=20 expression---but the notion of P-algebra for an arbitrary=20 PROP no longer makes sense. The moral is that a PROP in general may be built from=20 components which originate "in algebra" and components which=20 originate "in coalgebra"; or, indeed, from components which=20 originate in neither. It is, I think, only when all=20 components originate "in algebra"---which is to say that the=20 PROP is an operad---that the notion of P-monoidal functor is=20 mathematically sensible. However, all is not lost; for many of the PROPs of interest=20 are not just PROPs, but instances of some smaller notion=20 which allows "algebraic" and "coalgebraic" components=20 interacting according to some particular discipline. For=20 example, there is Gan's notion of dioperad, which is=20 essentially that of a one-object polycategory. The PROP for a=20 Frobenius algebra is an example of such a dioperad. Now, we=20 may speak of models for a dioperad in any polycategory; and=20 in particular, the functor category [C,D] between two=20 monoidal categories bears such a polycategorical structure,=20 wherein a model for the dioperad for Frobenius algebras is=20 precisely a Frobenius monoidal functor. Jeff Egger has=20 studied circumstances under which this polycategorical=20 structure of [C,D] is representable, and almost certainly=20 discusses this example of Frobenius monoidal functors; but he=20 is much more qualified than me to speak on this! Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]