From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8363 Path: news.gmane.org!not-for-mail From: Tarmo Uustalu Newsgroups: gmane.science.mathematics.categories Subject: Re: non-unital monads Date: Tue, 21 Oct 2014 00:02:09 +0300 (EEST) Message-ID: References: Reply-To: Tarmo Uustalu NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1413841003 5691 80.91.229.3 (20 Oct 2014 21:36:43 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 20 Oct 2014 21:36:43 +0000 (UTC) Cc: categories@mta.ca To: Vladimir Voevodsky Original-X-From: majordomo@mlist.mta.ca Mon Oct 20 23:36:37 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 1XgKd3-0007Re-51 for gsmc-categories@m.gmane.org; Mon, 20 Oct 2014 23:36:37 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:54386) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XgKch-0006KJ-UV; Mon, 20 Oct 2014 18:36:15 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XgKci-0006AD-Pu for categories-list@mlist.mta.ca; Mon, 20 Oct 2014 18:36:16 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8363 Archived-At: Dear Vladimir, 1. The answer to the first question is no, there can only be one unit for a given underlying functor and multiplication. (But for a given underlying functor and unit, there can of course be multiple multiplications.) 2. "Non-unital monads" are not difficult to find. On Set, you can consider, for example, - T X = X x S where (S, *) is some semigroup ass X x * mu_X = (X x S) x S -------> X x (S x S) ----- -> X x S The simplest special case is given by right zero semigroups: Take any set S and define s * s' = s'; one gets fst x S mu_X = (X x S) x S -------> X x S (For S with 2 or more elements, there is no unit.) - T X = lists over X of length at least n, for some fixed n mu_X = flattening of a list of lists into a list (For n \geq 2, there is no unit.) - For an endofunctor F, the free non-unital monad on F would be F^+ X = F (F^* X) \cong F^* (F X) where F^* is the free monad on F (assuming this exists). So concretely you can construct F+ in terms of initial algebras by F^+ X = F (mu Z. X + F Z) \cong mu Z. F X + F Z (for comparison, F^* X \cong mu Z. X + F Z) The free non-unital monad exists precisely when the free monad does, as you also have F^* X \cong X + F^+ X For your example, F X = X x X, one gets that F X is the set of all composite terms over variables from X, for a signature with one binary operation. (And free would mean left adjoint to forgetful as usual.) Kind regards, Tarmo U 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/ ]