From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7626 Path: news.gmane.org!not-for-mail From: Jeff Egger Newsgroups: gmane.science.mathematics.categories Subject: Re: Question on (co)monoids Date: Fri, 15 Mar 2013 07:22:26 -0700 (PDT) Message-ID: References: <1363096695.61914.YahooMailClassic@web172502.mail.ir2.yahoo.com> Reply-To: Jeff Egger NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1363505031 32598 80.91.229.3 (17 Mar 2013 07:23:51 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 17 Mar 2013 07:23:51 +0000 (UTC) To: claudio pisani , "categories@mta.ca" Original-X-From: majordomo@mlist.mta.ca Sun Mar 17 08:24:17 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1UH7x1-00070z-Jh for gsmc-categories@m.gmane.org; Sun, 17 Mar 2013 08:24:15 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:32833) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1UH7uf-00066i-6Y; Sun, 17 Mar 2013 04:21:49 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1UH7ug-0004G2-V0 for categories-list@mlist.mta.ca; Sun, 17 Mar 2013 04:21:50 -0300 In-Reply-To: <1363096695.61914.YahooMailClassic@web172502.mail.ir2.yahoo.com> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7626 Archived-At: Hi Claudio,=A0=0A=0AI was thinking yesterday night about the problem you po= sed,=A0=0A=0A> If C is a symmetrical monoidal category and every object has= a natural monoid=A0=0A> structure (that is any map is a monoid=A0 morphism= ) then C is cocartesian monoidal=A0=0A> (tensor =3D sums).=0A=0A=0Ain light= of the=A0three very different answers which arrived to it on=A0=0AWednesda= y. =A0I was sufficiently surprised by my own conclusions that=A0=0AI feel a= =A0need to share=A0them.=0A=0A=0AThe statement, as given, is a little vague= ; I interpret it as follows.=0A=0APROPOSITION 0: Let (V,@,I) be a symmetric= monoidal category, and=0Asuppose that the forgetful functor U : Mon(V,@,I)= --->V is split epi in=0Athe (1-)category of (mere) categories and functors.= =A0Then I is initial=0Aand @ is cocartesian.=0A=0A[A splitting of U maps e= ach object of V to a specific monoid,=0AA |-> (A,m_A,e_A), and every arrow = to itself. =A0In other words, every=0Aarrow f:A-->B must be a homomorphism = with respect to the specific=A0=0Astructures (A,m_A,e_A) and (B,m_B,e_B).= =0A=0AIt turns out, to my surprise, that Proposition 0 is false: I will=0Am= omentarily demonstrate this by means of a concrete counter-example;=0Abut f= irst let me address the issue of what is true.=0A=0AChris Heunen suggests p= atching the statement of=A0Proposition 0 as follows.=0A=0APROPOSITION 1: Le= t (V,@,I) be a symmetric monoidal category, and=0Asuppose that the forgetfu= l functor U : Mon(V,@,I)--->V is invertible=0Ain the (1-)category of (mere)= categories and functors. =A0Then I is=0Ainitial and @ is cocartesian.=0A= =0AProposition 1 is indeed true, but its utility is somewhat suspect.=0ATo = show that U is invertible one must show not only that every object=0AA carr= ies a monoid structure, but that every monoid structure on A=0Aequals the g= iven one.=0A=0AMike Shulman suggests instead (what amounts to):=A0=0A=0APRO= POSITION 2: Let (V,@,I) be a symmetric monoidal category, and=0Asuppose tha= t the forgetful functor U : Mon(V,@,I)--->V is split epi in=0Athe (1-)categ= ory of tensor categories and functors. =A0Then I is initial=0Aand @ is coca= rtesian.=0A=0ANot only is Proposition 2 true, but its truth implies that of= =0AProposition 1: the hypothesis of Proposition 2 requires that m_I and=0Ae= _I be the canonical isos, and that m_{A@B} and e_{A@B} be related to=0Am_A@= m_B and e_A@e_B in the usual way; namely, via the canonical isos=0AA@A@B@B-= ->A@B@A@B and I-->I@I, respectively. =A0This evidently follows=0Afrom the h= ypothesis of Proposition 1.=0A=0ABut it turns out that the hypothesis of Pr= oposition 2 is still much=0Astronger than is required. =A0For instance, tho= se parts which refer to=0Am_I, e_I, and e_{A@B} are entirely superfluous, a= nd the part referring=0Ato m_{A@B} is only used to establish the following = property, which=0Adoes not even refer to the symmetry of (V,@,I).=0A=0A(*) = id_A@e_B@e_A@id_B splits m_{A@B}=0A(modulo the canonical iso A@B-->A@I@I@B)= =0A=0AHence we arrive at the following.=0A=0APROPOSITION 3: Let (V,@,I) be = a monoidal category, and suppose that=0Athe forgetful functor U : Mon(V,@,I= )--->V is split epi in the=0A(1-)category of (mere) categories and functors= . =A0If (*) holds, then I=0Ais initial and @ is cocartesian.=0A=0A=0ANow I = will not write out a proof of Proposition 3, which is a tedious=0Aexercise = known (at least in spirit) to many. =A0But I will demonstrate=0Athe necessi= ty of (*) by means of the example promised above.=0A=0ALet V=3DSet, I=3D0, = and @ be the tensor product defined by=0A=A0=A0A@B =3D A + B + AxB.=0AThis = ``unusual'' symmetric monoidal structure on Set was discussed on=0Athe list= a few years ago in a thread initiated by Peter Selinger.=0A=0AI don't reme= mber whether it was mentioned at the time, but monoids in=0A(Set,@,I) are t= he same thing as semigroups in (Set,x)---i.e.,=0Asemigroups in the most ord= inary sense of the word. =A0For if m : AxA-->A=0Ais associative, then so is= [id_A,id_A,m] : A@A-->A. =A0Moreover, the=0Aunique map 0-->A is indeed a u= nit for any map A@A-->A of the form=0A[id_A,id_A,f]. =A0Conversely, every m= onoid in (Set,@,0) is of this form.=0A=0AIn this manner, we obtain an isomo= rphism between Mon(Set,@,I) and Sgp=0Ain the 1-category of mere categories,= which, moreover, commutes with=0Athe two ``underlying set'' functors. =A0B= ut U:Sgp-->Set is a split epi=0Athat is not invertible; for instance, one h= as the ``left band =A0 =A0 =A0 =A0 =A0=A0=0Afunctor'' Set-->Sgp which assig= ns to each set A, the semigroup (A,p_l)=0Awith p_l(a,b)=3Da. =A0(There is, = of course, also a ``right band functor''=0Awhich also splits U.)=0A=0AThis = is the promised counter-example to Proposition 0. =A0It is not a=0Acounter-= example to Proposition 3, however, because the left band=0Afunctor violates= (*). =A0Let f_{A,B} denote the endomorphism of A@B=0Adefined by composing = the following three arrows:=0A=A0=A0the canonical iso A@B-->A@I@I@B=0A=A0= =A0id_A@e_B@e_A@id_B=0A=A0=A0m_{A@B}=0A---then f_{A,B} is a non-trivial ide= mpotent on A@B (not the identity,=0Aas demanded by (*)). =A0Specificially, = it maps a pair (a,b) in third=0Asummand of A@B to its first component a in = the first summand of A@B.=0ANote that A+B is the split of this idempotent.= =0A(Obviously, the right band functor also violates (*).)=0A=0AThis situati= on is typical: in fact, it is easy to show that the maps=0Af_{A,B}, as defi= ned above, are always idempotents; moreover, the=0Afollowing generalisation= of Proposition 3 also holds.=0A=0APROPOSITION 4: Let (V,@,I) be a monoidal= category, and suppose that=0Athe forgetful functor U : Mon(V,@,I)--->V is = split epi in the=0A(1-)category of (mere) categories and functors. =A0Then = I is initial,=0Aand if each of the idempotents f_{A,B} is split by some obj= ect=0AS_{A,B}, then V has coproducts given by S_{A,B}.=0A=0AIn general, I g= uess a monoidal category (V,@,I) for which the=0Aforgetful functor U : Mon(= V,@,I)--->V is split epi is what a computer=0Ascientist might call a ``mode= l of sum types without beta-reduction''?=0AI.e., there are maps fst : A-->A= @B and snd : B-->A@B and a copairing=0Aoperation [,] satisfying fst[a,b]=3D= a, snd[a,b]=3Db, but not generally=0A[fst c,snd c]=3Dc.=0A=0AThat is all.= =0A=0ACheers,=0AJeff.=0A [For admin and other information see: http://www.mta.ca/~cat-dist/ ]