Date: Mon, 7 Jul 1997 09:22:35 -0300 (ADT) Subject: actions of groupes on categories Date: Sun, 6 Jul 1997 10:51:47 -0400 (EDT) From: James Stasheff In his recent Action du groupe des tresses sur une categorie Deligne considers more genrally an action of a monoid M (n'ayant pas necessairement d'unite')!! on a category C the major point is, of course, that the fucntion T: M --> Functors(C,C) satisfies Tf Tg is isomorphic to T(fg) not = but that the isomorphisms are STRICTLY coherent for fgh most of his attack is through generators and relations for speciifc examples of interest the pictures for the monoid of strictly positive braids are neat but hasn't this sort of thing been investigatged much more generally? and what about the case of weak coherence (cf. higher homotopies)? any references would be welcome ************************************************************ Until August 10, 1998, I am on leave from UNC and am at the University of Pennsylvania Jim Stasheff jds@math.upenn.edu 146 Woodland Dr Lansdale PA 19446 (215)822-6707 Jim Stasheff jds@math.unc.edu Math-UNC (919)-962-9607 Chapel Hill NC FAX:(919)-962-2568 27599-3250 Date: Tue, 8 Jul 1997 14:19:13 -0300 (ADT) Subject: Re: actions of groupes on categories Date: Tue, 8 Jul 1997 15:18:11 +1100 From: Ross Street Dear Jim I don't see the big deal about not having a unit in the monoid when looking at (pseudo-)actions; throw one in - we know how it should act. If you agree to this then what you are looking at is a pseudofunctor (or homomorphism of bicategories) R : M --> Cat where M is the monoid regarded as a one-object category. Indeed, M can be any category. There is an equivalence between the 2-category Hom(M,Cat) of homomorphisms from M to Cat and the 2-category Fib/M of opfibrations over M. The study of fibrations has gone off in many directions: for example, it provides the appropriate way of dealing with categories of all sizes (not just small) when working in a topos. As to higher homotopies, Cat doesn't really have enough dimensions for them. But there are trihomomorphisms M --> Bicat. More generally, there will be higher homomorphisms M --> WOC where WOC is the weak omega-category of weak omega-categories - someday - for now we have several fairly good definitions of the objects and arrows of WOC but that's as far as it goes. Higher fibrations is another interesting topic: Claudio Hermida knows about the 2-category case which is relevant to braids since 2-opfibrations over a 2-category M correspond to homomorphisms M --> 2-Cat where 2-Cat is self-enriched via the internal hom for the Gray tensor product of 2-categories (and it is in proving the coherence for this tensor product where braid groups first seriously entered category theory). Also, consider any braided monoidal bicategory B and let t be the n-th tensor power of some object of B. Then there is an action of the kind you describe of the n-string braid group on the hom-category B(t,t). But this is part of a longer story. Some References: John Gray, "Fibred and cofibred categories" Proc Conf Cat Alg, La Jolla 1965 (Springer 1966) [See references to Grothendieck's work in the Gray paper] John Gray, Formal Category Theory SLNM 391 (1974) John Gray, Coherence for the tensor product of 2-categories, and braid groups "Algebra, Topology, and Category Theory" (Academic Press 1976) 63-76 Jean Benabou, Introduction to bicategories SLNM 47 (1967) Jean Benabou, Fibrations petites et localement petites CR Acad Sci Paris A 281 (1975) 897-900 Benabou-Roubaud, Monades et descente CR Acad Sc Paris 270 (1970) 96-98 Gordon-Power-Street, Coherence for tricategories, Memoirs AMS #558 (Sept 1995) Day-Street, Monoidal bicategories and Hopf algebroids, Advances in Math (to appear; galley proofs returned) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Ross Street email: street@mpce.mq.edu.au Mathematics Department phone: +612 9850 8921 Macquarie University fax: +612 9850 8114 Sydney, NSW 2109 Australia Internet: http://www.mpce.mq.edu.au/~street/ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Date: Wed, 9 Jul 1997 10:22:52 -0300 (ADT) Subject: Re: actions of groupes on categories Date: Wed, 9 Jul 1997 08:14:31 -0400 (EDT) From: James Stasheff the Deligne paper Action du groupe des tresses... Inv math 128 (1997)159-175 ************************************************************ Until August 10, 1998, I am on leave from UNC and am at the University of Pennsylvania Jim Stasheff jds@math.upenn.edu 146 Woodland Dr Lansdale PA 19446 (215)822-6707 Jim Stasheff jds@math.unc.edu Math-UNC (919)-962-9607 Chapel Hill NC FAX:(919)-962-2568 27599-3250 Date: Wed, 9 Jul 1997 10:21:17 -0300 (ADT) Subject: Re: actions of groupes on categories Date: Wed, 9 Jul 1997 12:11:32 +1000 From: Michael Batanin An addition to Ross Street's answer. Wilst the notion of weak \omega-functor is not yet worked out adequately the corresponding topological theory of A_{\infty} and E_{\infty} maps between A_{\infty} and E_{\infty}-spaces has been constructed by Boardman,Vogt, May, Segal and others. The theory of natural transformations up to ALL higher homotopies between simplicial functors has been developed by Dawer,Kan, Cordier, Porter, Bourn, Batanin, Heller and others. Simplicial A_{\infty}-categories and A_{\infty}-functors were defined and studied by Batanin and topological version of it by Schwanzl and Vogt (they call them \Delta-categories and use the idea related to the Segal delooping mashine.) Some (not all) references: 1. Batanin M.A., Coherent categories with respect to monads and coherent prohomotopy theory, Cahiers Topologie et Geom. Diff., vol.XXXIV-4, pp.279-304, 1993. 2. Batanin M.A., Homotopy coherent category theory and A_{\infty}-structures in monoidal categories, to appear, dvi file available at http://www-math.mpce.mq.edu.au/~mbatanin/papers.html 3. Boardman J.M., R.M.Vogt, Homotopy Invariant Algebraic Structures on Topological Spaces, Lecture Notes in Math., vol. 347, Springer-Verlag, Berlin, Heidelberg, New York, 1973. 4. Cordier J.-M., Porter T., Vogt's Theorem on categories of homotopy coherent diagrams, Math. Proc. Cambridge Phil. Soc., vol. 100, pp 65-90, 1986. 5. Cordier J.-M., Porter T., Maps between homotopy coherent diagrams, Topology and its Appl., 28, pp.255-275, 1988. 6. Cordier J.-M., Porter T., Homotopy coherent category theory, to appear in Transactions of the AMS, 7. Dwyer W.G., Kan D.M., Realizing diagrams in the homotopy category by means of diagrams of simplicial sets, Proc. Amer. Math. Soc., 91, pp.456-460, 1984. 8.Heller A., Homotopy in Functor Categories, Transactions AMS., v.272, pp.185-202, 1982. 10. Schw\"{a}nzl R., Vogt R., Homotopy homomorphisms and the Hammock localization, Boletin de la Soc. Mat. Mexicana, 37, 1-2, pp.431-449, 1992. Date: Thu, 10 Jul 1997 16:49:45 +0200 (MET DST) Subject: Re: actions of groupes on categories Date: Wed, 9 Jul 1997 11:05:02 -0400 (EDT) From: James Stasheff for A_\infty cats and/or fucntors see also Fukawa where the applicaitons are to Floer cohomology if I recall and thence to physics! ************************************************************ Until August 10, 1998, I am on leave from UNC and am at the University of Pennsylvania Jim Stasheff jds@math.upenn.edu 146 Woodland Dr Lansdale PA 19446 (215)822-6707 Jim Stasheff jds@math.unc.edu Math-UNC (919)-962-9607 Chapel Hill NC FAX:(919)-962-2568 27599-3250