From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5435 Path: news.gmane.org!not-for-mail From: Mark Weber Newsgroups: gmane.science.mathematics.categories Subject: Re: Quantum computation and categories Date: Tue, 29 Dec 2009 15:33:51 +0100 Message-ID: References: Reply-To: Mark Weber NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: ger.gmane.org 1262182292 11608 80.91.229.12 (30 Dec 2009 14:11:32 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 30 Dec 2009 14:11:32 +0000 (UTC) To: John Baez , categories@mta.ca Original-X-From: categories@mta.ca Wed Dec 30 15:11:25 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 1NPzE7-0007Ik-Uc for gsmc-categories@m.gmane.org; Wed, 30 Dec 2009 15:08:40 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NPyr4-0001Zh-70 for categories-list@mta.ca; Wed, 30 Dec 2009 09:44:50 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5435 Archived-At: Greetings all and thanks to John for the friendly provocation ... I was very interested at Mark Weber's reaction to this problem. He said, > roughly, "So dagger-categories aren't really categories with extra > structure. Okay: they're something else! And that's fine." (I'd be happy > for him to correct my rough summary and make his point more precisely.) > OK. First of all when thinking about cobordisms, or paths, or homotopies, the act of reversing orientation is fundamentally strictly involutive. So at first glance, it's not at all clear what would be gained by weakening in this case, despite the light that replacing equations by isomorphisms sheds on so many other mathematical situations. Algebraically also, it appears natural to think of the strict involutions as more fundamental. Just as categories are algebras of a very nice and easily described monad on the presheaf topos of graphs, "dagger categories" (I think the terminology "involutive category" would be better) are algebras of an analogous monad on the presheaf topos of involutive graphs. An involutive graph is a graph together with an involution on the set of edges which switches sources and targets. Formally the dagger category monad is obtained by a canonical lifting of the category monad through the forgetful functor from involutive graphs to graphs, so we have a canonical monad distributive law describing this situation. The observations of the previous paragraph generalise a lot and rather easily, and this encourages me to resist any urge to weaken the involutory aspects of the notion of dagger category. So the "something else" of John's post is that dagger categories are *involutive graphs* with structure, and their higher dimensional analogues are *involutive n-globular sets* with structure. So all the way up the dimensional ladder, regardless of how weak your higher compositions happen to be, if involutions (to model orientation reversals) are to be part of the picture, then I think they should be strict and strictly compatible with all the compositions and coherence data. For those still interested, I'll now give a few more details. First some background. In http://arxiv.org/abs/0909.4715 Michael Batanin, Denis-Charles Cisinski and I reformulated much of the "Batanin approach" to defining higher categories as the study of monads on categories of enriched graphs, particularly those that arise from multitensors. Briefly, given a category V, one can associate to any "distributive multitensor on V" (which is a lax monoidal structure on V such that the n-ary tensor product functor V^n-->V preserves coproducts in each variable), a monad on the category GV of graphs enriched in V. So for example this process takes the cartesian product for Set to the category monad on Graph. The operads used by Batanin to define weak higher categories, seen as certain monads on the presheaf topos G^n(Set) of n-globular sets, also arise in this way. One can also consider the category G_i(V) of involutive graphs enriched in V, and so begin to consider structures defined by monads on the presheaf topos (G_i)^n(Set) of what would be sensible to call "involutive n-globular sets". An involutive graph enriched in V is a V-graph X together with, for each pair of objects a,b from X, maps i_(a,b) : X(a,b) --> X(b,a) in V such that for all a,b, i_(b,a)i_(a,b) = identity. It is easy to verify that both processes V |-> GV and V |-> G_i(V) preserve presheaf toposes, so (G_i)^n(Set) really is a presheaf topos. To spell out the generalisation alluded to above, let E be a distributive multitensor on V, and write (as in the above paper) Gamma(E) for its associated monad on GV -- corollary(4.5) of our paper indicates an explicit formula. This formula is easily adapted to the involutive case to describe the monad Gamma_i(E) on G_i(V) and this is by definition a canonical lifting of Gamma(E) though the forgetful G_i(V)-->GV. In summary, for any higher categorical structure of interest, there is an involutive version (eg one can define involutive Gray categories), and from the above remarks we understand as much about the monads which describe them as we do their non-involutive counterparts, and moreover there's a canonical distributive law relating them. For me the interesting question now is how to adapt this to give an explicit description of monads which describe weak higher groupoids (with strictly involutive "inverse operations"). Steve Lack and I observed recently that ordinary groupoids are algebras for a monad on the category of involutive graphs, which arises via a *weak* distributive law in the sense of http://www.tac.mta.ca/tac/volumes/22/12/22-12abs.html between the category monad on Graph and the involutive graph monad on Graph, but I really don't see yet how this generalises. Best new years wishes to all, Mark Weber [For admin and other information see: http://www.mta.ca/~cat-dist/ ]