From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: categories: Re: adjunction of symmetric monoidal closed categories Date: Wed, 28 Jan 2009 09:56:28 +1100 Message-ID: Reply-To: Steve Lack NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1233106481 14564 80.91.229.12 (28 Jan 2009 01:34:41 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 28 Jan 2009 01:34:41 +0000 (UTC) To: Bockermann Bockermann , categories Original-X-From: categories@mta.ca Wed Jan 28 02:35:54 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1LRzLL-00032X-Ms for gsmc-categories@m.gmane.org; Wed, 28 Jan 2009 02:35:51 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LRyfj-0005Z3-GU for categories-list@mta.ca; Tue, 27 Jan 2009 20:52:51 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:10 Archived-At: On 20/01/09 6:11 AM, "Bockermann Bockermann" wrote: > Dear mathematicians, > I wonder if the following is true. Has anybody a reference, if this is > the case? > > Let V and W be two complete and cocomplete symmetric monoidal closed > categories and > L: V <--> W :R > an adjunction of (lax) symmetric monoidal functors. Let D be a small V- > category. > Is it true that there is a V-isomorphism > V-Fun(D,RW) = R(W-Fun(LD,W)) ? > > (If not, is this at least the case if L is strict symmetric monoidal?) > > Thank you for any help. > Tony > > Dear Tony, I pointed out this fact in my reply (see below) to one of your earlier questions. In fact you don't need symmetry, and L is automatically strong monoidal. Regards, Steve Lack. %%%%% On 6/12/08 10:21 AM, "Bockermann Bockermann" wrote: > Dear mathematicians, > > could anybody give me a hint if the following assertion is true? > Let V be a complete and co-complete symmetric monoidal closed category. The > category sV of simplicial objects in V is also complete and co-complete > symmetric monoidal closed with the pointwise tensor. There is a V-adjunction > D:V<-->sV:Z > of the V-functor Z which evaluates in 0 and the discrete V-functor D. Does > this induce a V-Isomorphism of V-categories > V-Fun(K,ZC)~sV-Fun(DK,C) > for any small V-category K and any sV-category C? > > Please note that a similar statement is true for the non-enriched case [e.g. > Borceux2, Proposition 6.4.8.]. > > Thank you for any help. > > Tony > > Dear Tony, Yes, it is true. More generally, let F-|U:W-->V be a monoidal adunction. This means that V and W are monoidal categories, F and U are monoidal functors, monoidal natural transformations 1-->UF and FU-->1 satisfying the triangle equations. (A monoidal functor F:V-->W involves maps FX\otimes FY-->F(X\otimes Y) and I_W-->F(I_V), not necessarily invertible, but satisfying coherence conditions. In a monoidal adjunction, as above, the monoidal functor F is necessarily strong, so that the comparison maps are invertible. The comparison maps for U need not be invertible.) For a small V-category K and a W-category C we do indeed have an isomorphism V-Fun(K,UC) = U(W-Fun(FK,C)) of V-categories. I'll do my best to explain this via ascii. V-functors from K to UC are in bijection with W-functors from FK to C; this takes care of the object-part. For V-functors M,N:K-->UC, the hom-object V-Fun(K,UC)(M,N) is the equalizer of the evident maps ---> Pi_k UC(Mk,Nk) ---> Pi_{k,l} [K(k,l), UC(Mk,Nl)] in V, where the products run over all objects k and l of K. On the other hand, U(W-Fun(FK,C)(M,N)) is given by the equalizer of --> U(Pi_k C(Mk,Nk)) --> U Pi_{k,l} [FK(k,l),C(Mk,Nl)] or equivalently, since U is a left adjoint, the equalizer of --> Pi_k UC(Mk,Nk) --> Pi_{k,l} U[FK(k,l),C(Mk,Nl)] So we are now left to prove Lemma: U[FX,Y]=[X,UY], for X in V and Y in W. Proof: V(Z,U[FX,Y]) = W(FZ,[FX,Y]) = W(FZ\otimes FX,Y) = W(F(Z\otimes X),Y) = V(Z\otimes X,UY) = V(Z,[X,UY]) naturally in Z and so U[FX,Y]=[X,UY] as required. Regards, Steve Lack.