From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2425 Path: news.gmane.org!not-for-mail From: rjwood@mathstat.dal.ca (RJ Wood) Newsgroups: gmane.science.mathematics.categories Subject: Re: module for a category Date: Wed, 27 Aug 2003 11:11:07 -0300 (ADT) Message-ID: <20030827141108.1237B73667@chase.mathstat.dal.ca> References: <20030825175543Z10615-24564+241@calvin.math.vt.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018651 4106 80.91.229.2 (29 Apr 2009 15:24:11 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:24:11 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Aug 27 13:32:08 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 27 Aug 2003 13:32:08 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19s3BD-0002Tu-00 for categories-list@mta.ca; Wed, 27 Aug 2003 13:29:55 -0300 In-Reply-To: <20030825175543Z10615-24564+241@calvin.math.vt.edu> from "Stefan Forcey" at Aug 25, 2003 01:55:41 PM X-Mailer: ELM [version 2.5 PL2] X-Virus-Scanned: by AMaViS 0.3.12 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 17 Original-Lines: 95 Xref: news.gmane.org gmane.science.mathematics.categories:2425 Archived-At: Here is another twist on this circle of ideas which appeared in the introductory chapter of my 1976 thesis. Robin Cockett and I are working on a redevelopment of it. For monoidal (V,\ten, i), (promonoidal V will suffice) consider Brian Day's convolution (closed) monoidal structure on set^{V^op}. If A is a set^{V^op} category, it is helpful to think of A(-,-):A^op x A ---> set^{V^op} as A(-,-,-):A^op x V^op x A ---> set with the interpretation that A(a,v,b) provides a set of `v-indexed families' of arrows from a to b. The composite of a v-indexed family (v;f):a--->b with a w-indexed family (w;g):b--->c is a w\ten v family (w\ten v;gf):a--->c. Of course it may happen that for each a,b in A, A(a,-,b) is representable, by an object A[a,b] in V. In this case each (v;f):a--->b takes the form f:v--->A[a,b]. If for each a in A, and each v in V, A(a,v,-) is representable, by an object a.v in A, then the (v;f):a--->b take the form f:a.v--->b. Note that the identity a.v--->a.v considered as a v-indexed family (v,j):a--->a.v can be construed as a family of `sum-injections' for the `multiple' a.v. (Asking for a representing object {v,b} for A(-,v,b) leads to dual considerations.) Simultaneous representability in a,b and a,v is equivalent to the notion of `tensored V-category' mentioned below. In part this work was motivated by questions raised by Linton in `The multilinear Yoneda lemmas' SLN 195, 209--229, and also pursued by Reynolds in his 1973 Wesleyan thesis. For example, if A is a V-category and M is a V-actegory, in the nomenclature below, what is a V-functor A--->M, a V-functor M--->A? The familial approach, suggested by the 1970s work of Benabou, Pare/Schumacher, Rosebrugh and others provides a straightforward intuitive answer. For general set^{V^op}-categories A and M, the data for a set^{V^op}-functor F:A--->M sends, for each v in V, each v-indexed family (v;f):a--->b to a v-indexed family (v;Ff):Fa--->Fb. Each representability possibility for A and B allows for a compact presentation of the data. When A is a V-category then it suffices to know F on the generic families g:A[a,b]--->A[a,b]. In other words, one requires (A[a,b];Fg):Fa--->Fb. If M is also a V-category then Fg is what is usually denoted F_{a,b}:A[a,b]--->M[Fa,Fb], the effect of F on homs, but if M is a V-actegory it will take the form Fa.A[a,b]--->Fb. If A is a V-actegory then it suffices to know F on the generic (v,j):a--->a.v. For M a V-category we have Fj:v--->M[Fa,F(a.v)], while for M also a V-actegory we have Fj:Fa.v--->F(a.v), a form called `tensorial strength' by Anders Kock in a seeries of papers about mononoidal monads. In fact the 3x3 possibilities for `strengths' can be tabulated easily using these considerations: Write 1) for `powers' {v,b}, 2) for homs [a,b] and 3) for `multiples' a.v. Then the i,j th entry below provides the form of strength for a set^{V^op}-functor F:A--->M where A is of type i) and M is of type j) 1) 2) 3) 1) F{v,b}--->{v,Fb} v--->M[F{v,b},Fb] F{v,b}.v--->Fb 2) Fa--->{A[a,b],Fb} A[a,b]--->M[Fa,Fb] Fa.A[a,b]--->Fb 3) Fa--->{v,F(a.v)} v--->M[Fa,F(a.v)] Fa.v--->F(a.v) Best regards RJ Wood > What you are looking for may be similar to something I queried Ross Street in regard to earlier this summer. > I'll save him some time by putting here the relevant part of his response. > > > I think the one you first > >mention is what we have been calling V-actegories. Benabou looked at > >these rather than (as well as?) V-categories in the early days of > >monoidal categories. Pareigis also made use of them. More recently, > >publications of Paddy McCrudden involve them. There is a close > >connection with V-categories. A V-module V x A --> A in this sense > >for which we have a parametrized adjoint V(x,[a,b]) =~ A(x.a,b) > >makes A a V-category with V-valued hom [a,b]. > > > >Conversely, a tensored V-category becomes such a V-module. > > I recommend the work of McCrudden, who has developed among other things a > descent theoretic approach to the tensor product of V-actegories. > There is also resource in the work of Harald Lindner. > His paper, Enriched Categories and Enriched Modules, in Cahiers, Vol XXII-2 (1981) > develops morphisms between enriched categories and actegories, which he calls modules. > I'm curious about why it is that I have never seen his work referenced. > > Paul B Levy writes: > > > > Hi > > > > Is there a standard reference for the notion of "left module for a > > category"? (or right module, or bimodule) > > > > Is there any reference in the setting of ordinary categories rather than > > (or as well as) enriched categories or bicategories? > > > > Thanks > > Paul