From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8361 Path: news.gmane.org!not-for-mail From: Vladimir Voevodsky Newsgroups: gmane.science.mathematics.categories Subject: Re: non-unital monads Date: Mon, 20 Oct 2014 19:22:18 +0100 Message-ID: <0C628851-A81F-46AD-B200-D312A1A7922D@ias.edu> Reply-To: Vladimir Voevodsky NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Mac OS X Mail 8.0 \(1990.1\)) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1413840809 3582 80.91.229.3 (20 Oct 2014 21:33:29 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 20 Oct 2014 21:33:29 +0000 (UTC) Cc: "Prof. Vladimir Voevodsky" To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Mon Oct 20 23:33:23 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 1XgKZu-0005Xs-Sq for gsmc-categories@m.gmane.org; Mon, 20 Oct 2014 23:33:23 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:54372) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XgKZd-0005vx-Mj; Mon, 20 Oct 2014 18:33:05 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XgKZe-00063p-K2 for categories-list@mlist.mta.ca; Mon, 20 Oct 2014 18:33:06 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8361 Archived-At: Many thanks to everybody who answered my questions! I understand the picture with unitality being a property and not a = structure now. As for the universal characterization I have in mind something like = this: 1. For a functor F on a category with finite coproducts such that for = each X0 there exists the initial algebra I(F\coprod X0) of the functor X |-> F(X)\coprd X0, these initial = algebras are functorial=20 and in fact X |-> I(F\coprod X) has an obvious monad structure and this = monad is the free monad generated by F.=20 This construction is what connects free monads with free algebras. 2. What can one do for a non-unital monad? It seems to me at the moment = that the functor X |-> I(F\coprod F(X))=20 may be the free non-unital monad generated by F.=20 Vladimir. > On Oct 20, 2014, at 5:47 PM, Marek Zawadowski = wrote: >=20 > Hi, >=20 > 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 >=20 > Mon( End(C) ) ---> nuMon ( End(C) ) ---> End(C) >=20 > 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. >=20 > 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 >=20 > M. Barr, Coequalizers and Free Triples, Math. Z. 116, pp. 307-322 = (1970) >=20 > 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 >=20 > Theories of analytic monads. Math. Str. in Comp. Sci. pp. 1-33, (2014) >=20 > Monads of regular theories. Appl. Cat. Struct. pp. 9331-9364, (2013) >=20 > As Tom and Peter remarked, if a monoid has a left unit and a right = unit, > they need to be equal. >=20 > Best regards, > Marek >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]