Subject: Re: actions of groupes on categories

From: categories <[email-masked]>
Date: Tue, 8 Jul 1997 14:19:04 -0300 (ADT)

Date: Tue, 8 Jul 1997 15:18:11 +1100
From: Ross Street <[email-masked]>

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: [email-masked]
Mathematics Department         phone:      +612 9850 8921
Macquarie University             fax:      +612 9850 8114
Sydney, NSW 2109
Australia                   Internet: http://www.mpce.mq.edu.au/~street/
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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From: categories <[email-masked]>
Date: Wed, 9 Jul 1997 10:20:37 -0300 (ADT)

Date: Wed, 9 Jul 1997 12:11:32 +1000
From: Michael Batanin <[email-masked]>

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.

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From: categories <[email-masked]>
Date: Wed, 9 Jul 1997 10:22:18 -0300 (ADT)

Date: Wed, 9 Jul 1997 08:14:31 -0400 (EDT)
From: James Stasheff <[email-masked]>

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		[email-masked]

	146 Woodland Dr
        Lansdale PA 19446       (215)822-6707	

	Jim Stasheff		[email-masked]
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	27599-3250

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From: categories <[email-masked]>
Date: Thu, 10 Jul 1997 10:20:03 -0300 (ADT)

Date: Wed, 9 Jul 1997 11:05:02 -0400 (EDT)
From: James Stasheff <[email-masked]>

for A_\infty cats and/or fucntors see also Fukawa
where the applicaitons are to Floer cohomology if I recall
and thence to physics!

	146 Woodland Dr