From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8359 Path: news.gmane.org!not-for-mail From: Peter Johnstone Newsgroups: gmane.science.mathematics.categories Subject: Re: non-unital monads Date: Mon, 20 Oct 2014 10:31:04 +0100 (BST) Message-ID: References: Reply-To: Peter Johnstone NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1413800096 8834 80.91.229.3 (20 Oct 2014 10:14:56 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 20 Oct 2014 10:14:56 +0000 (UTC) Cc: categories@mta.ca To: Vladimir Voevodsky Original-X-From: majordomo@mlist.mta.ca Mon Oct 20 12:14:51 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 1Xg9zG-00015s-Ld for gsmc-categories@m.gmane.org; Mon, 20 Oct 2014 12:14:50 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:53992) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1Xg9yw-0002xv-4D; Mon, 20 Oct 2014 07:14:30 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Xg9yv-0005g0-RT for categories-list@mlist.mta.ca; Mon, 20 Oct 2014 07:14:29 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8359 Archived-At: The answer to Vladimir's first question is no. Suppose \mu: TT --> T has two units \eta, \zeta: 1 --> T. Then, for any A, the composite \mu_A.T\eta_A.\zeta_A reduces to \zeta_A by one unit law; but it's equal to \mu_A.\zeta_TA.\eta_A by naturality of \zeta, and this reduces to \eta_A by the other unit law. (If you don't demand that the units be `two-sided' then the answer is yes.) Peter Johnstone On Sat, 18 Oct 2014, Vladimir Voevodsky wrote: > 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/ ]