From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8687 Path: news.gmane.org!not-for-mail From: Claudio Hermida Newsgroups: gmane.science.mathematics.categories Subject: Re: coherence for symmetric monoidal and (co)affine categories Date: Sun, 23 Aug 2015 22:44:16 -0300 Message-ID: References: Reply-To: Claudio Hermida NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1440510117 1839 80.91.229.3 (25 Aug 2015 13:41:57 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 25 Aug 2015 13:41:57 +0000 (UTC) To: Paul Blain Levy , categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Tue Aug 25 15:41:50 2015 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.7.22]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1ZUETy-0006tS-6v for gsmc-categories@m.gmane.org; Tue, 25 Aug 2015 15:41:46 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:44468) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1ZUESM-0006tD-Ui; Tue, 25 Aug 2015 10:40:06 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ZUESN-0000tk-A0 for categories-list@mlist.mta.ca; Tue, 25 Aug 2015 10:40:07 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8687 Archived-At: Dear Paul, On 2015-08-22 11:06 AM, Paul Blain Levy wrote: > Dear all, > > Given a category C, define symm(C) to be the following category: > > - an object is a finite family [or finite sequence, if preferred] of > C-objects > > - a morphism from (C_i | i in I) to (D_j | j in J) consists of an > bijection f : I --> J and, for each i in I, a C-morphism C_i --> D_fi. > > Define coaff(C) likewise but with "injection" instead of "bijection". > > It seems to be folklore that > > (1) symm(C) is the free symmetric monoidal category on C > > (2) coaff(C) is the free coaffine category (symmetric monoidal category > with initial unit) on C. > > In the special case where C is discrete, these statements follow from > the coherence arguments in Mac Lane's "Natural associativity and > commutativity" and Petric's "Coherence in substructural categories". > > But for general C, where are these statements proved? > > Paul > > > The 'coaffine envelope' construction of [HT12, Cor 2.12] provides the free coaffine category 'D' over a symmetric one D (wrt strong symmetric functors). So combining this construction with Symm, one gets Coaff(C) = 'Symm(C)' The explicit construction in ibid does not spell out this special case, except for C =1, where one gets the familiar category of finite sets and injections (with sum as tensor). The general description yields Coaff(C) ( X, Y) = [ \Coprod_{W\in Symm(C)} Symm(C)(X \tensor W, Y)]_~ that is, morphisms f:X \tensor W -> Y, indexed by W, subject to an equivalence relation via directed paths. It reduces to the explicit description of Coaff(C) above by noting that every equivalence class has a unique underlying injection i:|X| -> |Y| between the underlying index sets, which gives a canonical representative of ~-classes with W = (|W|, {Y_w}_{w\in |W|}) where |W| is the complement of the image of i in |Y|. As for Symm(C), I am not positive about the original reference, but surely it is worthwhile referring to the classical [JS93] which gives an explicit description of the free braided monoidal category on a given category (Prop. 2.2(b) and Cor. 2.4). One easily gets Symm(C) by restricting to braidings which are symmetries (so the construction involves permutation groups rather braid groups). The general construction (a Grothendieck construction) is introduced in Kelly's theory of clubs [K74]. The general theory that encompasses these monadic constructions is that of *polynomial monads*, as in the recent [W15] (and references therein for this rich theory), which includes the construction of Symm among the many examples. Claudio REFERENCES [HT12] - Hermida, Claudio, and Robert D. Tennent. "Monoidal indeterminates and categories of possible worlds."/Theoretical Computer Science/430 (2012): 3-22. [JS93] - Joyal, Andr??, and Ross Street. "Braided tensor categories."/Advances in Mathematics/102.1 (1993): 20-78. [K74] - Kelly, Gregory Max. "On clubs and doctrines."/Category seminar/. Springer Berlin Heidelberg, 1974. [W15] - Weber, Mark. "Polynomials in categories with pullbacks."/Theory and Applications of Categories/30.16 (2015): 533-598. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]