From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5359 Path: news.gmane.org!not-for-mail From: Ross Tate Newsgroups: gmane.science.mathematics.categories Subject: Lax Monoidal Functor Terminology Date: Sat, 12 Dec 2009 18:41:22 -0800 Message-ID: Reply-To: Ross Tate NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: ger.gmane.org 1260746322 7504 80.91.229.12 (13 Dec 2009 23:18:42 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 13 Dec 2009 23:18:42 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Mon Dec 14 00:18:35 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 1NJxhg-000274-9J for gsmc-categories@m.gmane.org; Mon, 14 Dec 2009 00:18:16 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NJx9K-0004Js-RL for categories-list@mta.ca; Sun, 13 Dec 2009 18:42:46 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5359 Archived-At: Hello, I've identified some useful non-standard properties of lax monoidal functors on Cartesian-like monoidal categories, and I am curious as to whether these properties already have names. Suppose I have a monoidal category with diagonals (diag : A -> A * A). Some lax monoidal functors (F, merge: FA * FB -> F(A * B)) have the property that diag;merge : FA -> F(A * A) equals F(diag). Is there a name for this property for either the special Cartesian case or the general case? What about the 2-categorical case where there's a 2-cell from diag;merge to F(diag) or vice-versa? Now suppose I have a monoidal category with terminators (term : A -> I). Some lax monoidal functors (F, unit: I -> FI) have the property that term;unit : FA -> FI equals F(term). Again, is there a name for this for the Cartesian, general, or 2-categorical cases? While I'm at it, is there a name for Cartesian-like monoidal categories with diagonals, terminators, and projections? The best I've found is Cartesian structures on bicategories, but my monoidal categories do not have the required 2-cells/identities. At best, when there are 2-cells, the 2-cells are not isomorphisms, which I guess makes it a lax Cartesian structure. Thanks for any help you can provide, Ross P.S. This is my first time using this mailing list, so please tell me if there's anything I should know about. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]