Notes of two lectures

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Date: Thu, 24 Apr 1997 15:37:19 +1100
From: Ross Street <street@mpce.mq.edu.au>

This is to announce the placement on the WWW of the notes of my two
lectures at the Conference on Higher Category Theory and Mathematical
Physics, Northwestern University (Evanston, Illinois;  28-30 March 1997).
The site is:

       <www-math.mpce.mq.edu.au/~coact/street_nw97.ps>

[I have tried to eliminate offending fonts and to accommodate funny US
paper size. Thanks to Sjoerd Crans for helping here.]

Title: The role of Michael Batanin's monoidal globular categories
            Lecture I:  Globular categories and trees
            Lecture II: Higher operads and weak omega-categories

        This is a report on recent work of Michael Batanin. The goal of his
work is to provide an environment for defining the concepts associated with
weak omega-categories and for developing the ensuing theory.  The approach
is "globular".

        To put this in context, I might mention some important steps in the
development of weak omega-categories. Categories were defined by
Eilenberg-Mac Lane in 1945. Monoidal and symmetric monoidal categories were
defined by Mac Lane in 1963.  Ehresmann defined (strict) n-categories in
1966.  Bénabou defined bicategories in 1967.  In the early 80s, monoidal
bicategories were in the air but a full definition was not published in
that period. Joyal-Street defined braided monoidal categories in 1985.
Gordon-Power-Street defined tricategories in 1991 (this, and the coherence
theorem, were published in 1995). Braided monoidal categories were defined
by Kapranov-Voevodsky-Baez-Neuchl-Breen around 1993.  Trimble produced a
definition of tetracategory in 1995.
        Diverse approaches to weak n-categories for all n have appeared.
Street (1985) suggested a simplicial definition with horn filler
conditions.  Trimble (1994) approached the problem using operads and
Stasheff associahedra.  Baez-Dolan (1995) have a definition using typed
operads and opetopes.  Tamsamani (1996) gave a multisimplicial definition.
Batanin uses higher operads and globular sets.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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/
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