From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8360 Path: news.gmane.org!not-for-mail From: Marek Zawadowski Newsgroups: gmane.science.mathematics.categories Subject: Re: non-unital monads Date: Mon, 20 Oct 2014 18:47:49 +0200 Message-ID: References: Reply-To: Marek Zawadowski NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8; DelSp="Yes"; format="flowed" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1413840664 2135 80.91.229.3 (20 Oct 2014 21:31:04 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 20 Oct 2014 21:31:04 +0000 (UTC) Cc: categories@mta.ca To: Vladimir Voevodsky Original-X-From: majordomo@mlist.mta.ca Mon Oct 20 23:30:58 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.127]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1XgKXX-0004Fy-PC for gsmc-categories@m.gmane.org; Mon, 20 Oct 2014 23:30:55 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:54363) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XgKWu-0005mC-FU; Mon, 20 Oct 2014 18:30:16 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XgKWv-0005zN-LE for categories-list@mlist.mta.ca; Mon, 20 Oct 2014 18:30:17 -0300 In-Reply-To: Content-Disposition: inline Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8360 Archived-At: Hi, Monads on a category C are monoids in the strict monoidal category End(C) of endofunctors on C and natural transformations. We have the forgetful functors Mon( End(C) ) ---> nuMon ( End(C) ) ---> End(C) forgetting from monoids to non-unital monoids and then to endofunctors. These functors might have left adjoints. This answers the second question concerning universal properties. If C is Set, and we restrict objects in End(Set) to functors with rank at most m (for some cardinal m) , then it was shown in M. Barr, Coequalizers and Free Triples, Math. Z. 116, pp. 307-322 (1970) that the left adjoint to the composition of the above functors exists giving rise to a monad for monads on End(Set) with rank at most m. There are also refinements of this result saying that the free monads on polynomial, analytic, and semi-analytic functors are polynomial, analytic, and semi-analytic, respectively. The first occurs in the unpunlished book of Joachim Kock and the last two in the papers I wrote recently with S. Szawiel Theories of analytic monads. Math. Str. in Comp. Sci. pp. 1-33, (2014) Monads of regular theories. Appl. Cat. Struct. pp. 9331-9364, (2013) As Tom and Peter remarked, if a monoid has a left unit and a right unit, they need to be equal. Best regards, Marek Cytowanie Vladimir Voevodsky : > Hello, > > I am trying to find some information about non-unital monads > (gadgets with \mu but without \eta). > > In particular I am interested in the following two questions: > > 1. Given a non-unital monad can it have two different "unitality" structures? > > 2. Is there a concept of a free non-unital monad? For example, I can think of > the "free" non-unital monad generated by the functor X |-> X^2 on > sets as the monad > that sends a set X into the set of "homogeneous" expressions made > with one binary operation > s such that there is s(x1,x2) and s(s(x1,x2),s(x3,x4)) but no x1 > itself and no s(x1,s(x2,x3)). > But what is the universal characterization of it? > > Thanks! > Vladimir. > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]