Notes of two lectures

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

Notes of two lectures

[categories]

Date: Wed, 30 Apr 1997 11:39:29 +1100
From: Ross Street <street@mpce.mq.edu.au>

With computers, things we expect to be simple never are!

Last week I announced the placement on the WWW of the notes (see the short
description below) 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'll spare you the details of the problems, but, up until today, what was
at this site was an old version prepared before the conference. This is not
what I had intended. I am truly sorry to people who have downloaded that
version already.

The correct version is NOW at the site. There still seems to be a problem
when the document is viewed (by some "ghost" technology) but it does (at
least for us) print out pretty well. There is a colour table on page 2
which even seems to view correctly!  I am extremely grateful to Ross Moore
and Sjoerd Crans for helping me out of the mess I (and my little Mac)
created.

I invite people who have downloaded the old version to try again; I am
really sorry.

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".  Apart from providing a precise definition of weak
omega-category, the work gives a new algebra of planar trees and uses them
to define higher operads which I believe will find many other applications.

Note that, in the meantime (and again not without trouble!), Michael
Batanin has announced his paper containing the full details of this part of
his work. It is available at:
        <http://www-math.mpce.mq.edu.au/~mbatanin/papers.html>

Happy surfing and enjoy the trees.

Ross