Symmetric monoidal closed categories
Dear mathematicians, I wonder if the following is true. Has anybody a reference, if this is the case? For a complete and co-complete symmetric monoidal closed category C and a small category D the functor category Fun(D,C) is pointwise a symmetric monoidal category. Is this a closed symmetric monoidal structure? This is true for simplicial sets and simplicial abelian groups for example. Thank you for any help. Tony
[From moderator: apologies to Vincent Schmitt. He posted the answer below and the second item which will be reposted. The correct From: field was inadvertently omitted.] On Nov 19 2008, Bockermann Bockermann wrote:
Dear mathematicians,
I wonder if the following is true. Has anybody a reference, if this is the case?
For a complete and co-complete symmetric monoidal closed category C and a small category D the functor category Fun(D,C) is pointwise a symmetric monoidal category. Is this a closed symmetric monoidal structure? This is true for simplicial sets and simplicial abelian groups for example.
Thank you for any help.
What is true is the following: if D,C are symmetric monoidal then the category of symmetric monoidal functors D->C admits a symmetric monoidal structure given pointwise by that of C. The crucial point is the symmetry.
Tony
On Nov 19 2008, Bockermann Bockermann wrote:
Dear mathematicians,
I wonder if the following is true. Has anybody a reference, if this is the case?
For a complete and co-complete symmetric monoidal closed category C and a small category D the functor category Fun(D,C) is pointwise a symmetric monoidal category. Is this a closed symmetric monoidal structure? This is true for simplicial sets and simplicial abelian groups for example.
Thank you for any help.
What is true is the following: if D,C are symmetric monoidal then the category of symmetric monoidal functors D->C admits a symmetric monoidal structure given pointwise by that of C. The crucial point is the symmetry.
Tony
[From moderator: apologies to Vincent Schmitt. He posted the answer below and the first item recently reposted. The correct From: field was inadvertently omitted.] A bit more. You may be interested by the V-category of V-functors [A,B] for V-categories A and B -- Take care of the sizes though. V= SSet, Ab etc... References for this: Day and Kelly certainly. Kelly's "Basic concepts of enriched category theory" or Day's thesis and early papers. Best, V. On Nov 19 2008, Bockermann Bockermann wrote:
Dear mathematicians,
I wonder if the following is true. Has anybody a reference, if this is the case?
For a complete and co-complete symmetric monoidal closed category C and a small category D the functor category Fun(D,C) is pointwise a symmetric monoidal category. Is this a closed symmetric monoidal structure? This is true for simplicial sets and simplicial abelian groups for example.
Thank you for any help.
Tony
A bit more. You may be interested by the V-category of V-functors [A,B] for V-categories A and B -- Take care of the sizes though. V= SSet, Ab etc... References for this: Day and Kelly certainly. Kelly's "Basic concepts of enriched category theory" or Day's thesis and early papers. Best, V. On Nov 19 2008, Bockermann Bockermann wrote:
Dear mathematicians,
I wonder if the following is true. Has anybody a reference, if this is the case?
For a complete and co-complete symmetric monoidal closed category C and a small category D the functor category Fun(D,C) is pointwise a symmetric monoidal category. Is this a closed symmetric monoidal structure? This is true for simplicial sets and simplicial abelian groups for example.
Thank you for any help.
Tony
Hi Bob, who is the mysterious "V"? It seems that the "From" field was completely erased from today's two messages on the categories list. -- Peter
From categories@mta.ca Wed Nov 19 11:49:10 2008 Return-Path: <categories@mta.ca> X-Spam-Checker-Version: SpamAssassin 3.2.3 (2007-08-08) on test4net.MathStat.Dal.Ca X-Spam-Level: * X-Spam-Status: No, score=2.0 required=4.8 tests=BAYES_00,DATE_IN_PAST_06_12, DCC_CHECK,DNS_FROM_SECURITYSAGE,LOCALPART_IN_SUBJECT autolearn=no version=3.2.3 X-Original-To: selinger@mathstat.dal.ca Delivered-To: selinger@mathstat.dal.ca Received: from sxt-sm-1.ucis.dal.ca (SXT-SM-1.UCIS.Dal.Ca [129.173.5.72]) by chase.mathstat.dal.ca (Postfix) with SMTP id C431C5C279 for <selinger@mathstat.dal.ca>; Wed, 19 Nov 2008 11:49:10 -0400 (AST) Received: from kil-av-3.ucis.dal.ca (KIL-AV-3.UCIS.Dal.Ca [129.173.1.208]) by sxt-sm-1.ucis.dal.ca (8.14.2+Sun/8.14.2) with SMTP id mAJFlK6R009915 for <selinger@mathstat.dal.ca>; Wed, 19 Nov 2008 11:47:20 -0400 (AST) Received: from (KIL-MX-1.UCIS.Dal.Ca [129.173.1.129]) by kil-av-3.ucis.dal.ca with smtp id 12cc_0077_595c4a3c_b651_11dd_886f_00142222ffad; Wed, 19 Nov 2008 11:47:19 -0400 X-Dal-Submitter-IP: 137.122.6.67 Received: from mx1.uottawa.ca (mx1.uottawa.ca [137.122.6.67]) by kil-mx-1.ucis.dal.ca (8.14.2+Sun/8.14.2) with ESMTP id mAJFjkFr023853 for <selinger@mathstat.dal.ca>; Wed, 19 Nov 2008 11:45:46 -0400 (AST) Received: from localhost (unknown [127.0.0.1]) by mx1.uottawa.ca (Postfix) with ESMTP id E7D2340092; Wed, 19 Nov 2008 15:45:40 +0000 (UTC) X-Virus-Scanned: amavisd-new at uottawa.ca Received: from mx3.uottawa.ca ([137.122.6.69]) by localhost (mx1.uottawa.ca [137.122.6.67]) (amavisd-new, port 10024) with ESMTP id 8x5OMUpfotAs; Wed, 19 Nov 2008 10:45:40 -0500 (EST) Received: from science.uottawa.ca (scimail.science.uottawa.ca [137.122.43.19]) by mx3.uottawa.ca (Postfix) with ESMTP id C1C47200027; Wed, 19 Nov 2008 10:45:40 -0500 (EST) Received: from Spooler by science.uottawa.ca (Mercury/32 v3.32) ID MO00F3DC; 19 Nov 08 10:45:40 -0500 Received: from spooler by science.uottawa.ca (Mercury/32 v3.32); 19 Nov 08 10:45:17 -0500 Received: from mx3.uottawa.ca (137.122.6.69) by science.uottawa.ca (Mercury/32 v3.32) with ESMTP ID MG00F3D8; 19 Nov 08 10:45:14 -0500 Received: from localhost (unknown [127.0.0.1]) by mx3.uottawa.ca (Postfix) with ESMTP id F05A520002A; Wed, 19 Nov 2008 15:45:13 +0000 (UTC) X-Virus-Scanned: amavisd-new at uottawa.ca Received: from mx2.uottawa.ca ([137.122.6.68]) by localhost (mx3.uottawa.ca [137.122.6.69]) (amavisd-new, port 10024) with ESMTP id 1nGCtGdbKSkY; Wed, 19 Nov 2008 10:45:07 -0500 (EST) Received: from mailserv.mta.ca (mailserv.mta.ca [138.73.1.1]) by mx2.uottawa.ca (Postfix) with ESMTP id 8137128090; Wed, 19 Nov 2008 10:45:07 -0500 (EST) Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from <categories@mta.ca>) id 1L2p8A-00028o-19 for categories-list@mta.ca; Wed, 19 Nov 2008 11:38:14 -0400 To: categories@mta.ca Subject: categories: Re: Symmetric monoidal closed categories Date: 19 Nov 2008 08:39:29 +0000 Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset=ISO-8859-1 Sender: categories@mta.ca Precedence: bulk Reply-To: Bob Rosebrugh <rrosebru@mta.ca> Message-Id: <E1L2p8A-00028o-19@mailserv.mta.ca> From: categories@mta.ca X-NAI-Spam-Flag: NO X-NAI-Spam-Threshold: 5 X-NAI-Spam-Score: 0 X-NAI-Spam-Version: 2.1.0.7694 : core <3150> : streams <160088> : uri <29793 X-Myspam-seen: Wed Nov 19 11:49:17 AST 2008 X-Myspam: no X-Myspam-reason: categories-list@mta.ca Status: O
A bit more. You may be interested by the V-category of V-functors [A,B] for V-categories A and B -- Take care of the sizes though. V= SSet, Ab etc... References for this: Day and Kelly certainly. Kelly's "Basic concepts of enriched category theory" or Day's thesis and early papers. Best, V. On Nov 19 2008, Bockermann Bockermann wrote:
Dear mathematicians,
I wonder if the following is true. Has anybody a reference, if this is the case?
For a complete and co-complete symmetric monoidal closed category C and a small category D the functor category Fun(D,C) is pointwise a symmetric monoidal category. Is this a closed symmetric monoidal structure? This is true for simplicial sets and simplicial abelian groups for example.
Thank you for any help.
Tony
participants (4)
-
Bockermann Bockermann -
categories＠mta.ca -
selinger -
vs27＠mcs.le.ac.uk