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Abstracts for the IMA 2004 Summer Program:
n-Categories: Foundations and Applications
June 7-18, 2004

Mathematics of Materials and Macromolecules: Multiple Scales, Disorder, and Singularities, September 2004 - June 2005

John Baez (Department of Mathematics, University of California at Riverside) baez@math.ucr.edu http://www.math.ucr.edu/home/baez/

Why n-Categories?
Slides:
pdf    ps

Research on n-categories is revolutionizing our concept of mathematics by teaching us how to think of every interesting equation as the summary of an interesting process. Here we sketch how this approach leads to a new understanding of even the simplest mathematical structures: in particular, of the natural numbers. A topological study of the origin of the natural number concept suggests that in the "true natural numbers", addition satisfies an infinite hierarchy of coherence laws for associativity, an infinite hierarchy of coherence laws for commutativity, together with an infinite hierarchy of further hierarchies of coherence laws. All these are built into the concept of the "free k-tuply monoidal n-category on one generator", and this should admit a description as the n-category of "n-braids in codimension k." The objects here are elements of the space of finite subsets of k-dimensional Euclidean space. The morphisms are paths in this space, the 2-morphisms are paths of paths in this space, and so on, with the n-morphisms being homotopy classes of paths of paths of paths... in this space. Similarly, the integers should be related to the n-category of "n-tangles in codimension k."

John Baez (Department of Mathematics, University of California at Riverside) baez@math.ucr.edu http://www.math.ucr.edu/home/baez/

What n-Categories Should Be Like
Slides:
pdf    ps

We describe features that any useful theory of n-categories should have. In particular, there should be three specially nice sorts of n-categories: k-tuply monoidal n-categories, n-groupoids and strict n-categories. We describe the effects of imposing these extra conditions separately and in combination, and conjecture the existence of a weakly commutative cube of free and forgetful (n+1)-functors relating the resulting eight classes of n-categories. Among other things, this cube would give an explanation of the relation between n-categories, homotopy theory, stable homotopy theory, and homology theory.

John Baez (Department of Mathematics, University of California at Riverside) baez@math.ucr.edu http://www.math.ucr.edu/home/baez/

Space and State, Spacetime and Process
Slides:   pdf    ps

General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two, which is best understood using category theory. General relativity describes space and spacetime in terms of objects and morphisms in nCob, the category of n-dimensional cobordisms. Quantum theory describes states and processes using objects and morphisms in Hilb, the category of Hilbert spaces. The analogy between general relativity and quantum theory is made precise by the fact that both nCob and Hilb are "symmetric monoidal categories with duals". Work on string theory and loop quantum gravity suggests that the analogy goes deeper, in a way that is best understood using n-categories. The "TQFT hypothesis" is a preliminary attempt to make this precise.

Eugenia Cheng (Department of Pure Mathematics & Mathematical Statistics, Cambridge University) elgc2@cam.ac.uk

Multicategories and Related Definitions and Opetopic Definitions

General: In these talks I intend to give a brief overview and then let the pace be set by the audience. I won't be producing detailed notes because the details will depend heavily on what the audience wants at the time. However, general notes can be found in Higher-Dimensional Categories: an illustrated guidebook, Chapters 2, 3 and 4. This is available from http://www.dpmms.cam.ac.uk/~elgc2/guidebook

Multicategories and related definitions: Tom Leinster will begin with an introduction to generalised multicategories and operads. I will then explain related definitions that use the notion of algebraic structure on underlying data consisting of "globular" cells. These definitions are expressed in the form: An $\omega$-category is an algebra for a certain monad/operad". I will explain the various components of the definition as necessary - I envisage this including magmas, contractions and the construction of the initial operads in question, possibly globular sets and probably not monads and algebras. (A crash course on monads and algebras can be found in the Guide Book, Section 2.4. I will be happy to talk about it informally beforehand if anyone needs it.)

Opetopic definitions: My aim in this talk will be to convey the idea of the opetopic definitions while avoiding as much of the technical detail as possible. Experience has shown that this is highly expedient for those wishing to get a feel for the theory without necessarily wanting to calculate with it. I will explain the ideas behind the theory and the various components of the definition: opetopes, opetopic sets, niches, and universal cells. I will show how to recover a classical bicategory from an opetopic 2-category as this tends to help shed a great deal of light on the various components of the definition. If there is time I will discuss various notions of strictness and points of comparison with other definitions.

Note that for the benefit of overall understanding I will aim to avoid talking in detail about: the use of multicategories in the iterative construction of opetopes, the different technical approaches to this construction, and the construction of morphisms of opetopes. I will be happy to talk about this in an informal session another day if anyone is keen to hear these details.

David Neil Corfield (Oxford University)

N-Category Theory as a Catalyst for Change in Philosophy
Paper:
pdf

Uli Fahrenberg (Department of Mathematical Sciences, Aalborg University, Denmark) uli@math.aau.dk http://www.math.auc.dk/~uli/getco04

A Dihomotopy Double Category of a Po-Space
Slides:   pdf    ps

We construct a double category on a po-space capturing dihomotopy information in dimension 2. A po-space is a topological space with a partial order defined on the points, and dihomotopy is the po-space analogy of the usual homotopy notion.

The 1-cells in our double category are dipaths modulo reparametrisation, and the squares are maps from the unit square into the po-space, modulo a notion of dihomotopy taking the dipath reparametrisations into account. We show that our dihomotopy double category obeys a van Kampen type property.

Thomas M. Fiore (Department of Mathematics, University of Michigan) fioret@umich.edu

Pseudo Algebraic Structures in Conformal Field Theory (impromptu talk)
Introduction:   dvi    pdf   ps
Slides:
dvi    pdf
ps

This talk deals with the pseudo algebraic structure of gluing and disjoint union on the category of rigged surfaces and its role in the definition of conformal field theory. Pseudo algebras over Lawvere theories and 2-theories are treated in order to capture the pseudo algebraic structure. This work is an application of weak 2-categorical concepts to physics.

Philippe Gaucher (Universit� Paris 7 Denis-Diderot) http://www.pps.jussieu.fr/~gaucher

Towards a Homotopy Theory of Higher Dimensional Automata
Slides:   pdf

We give an overlook of our work about globular complexes and flows. Globular complexes, flows, S-homotopy and T-homotopy are explained by examples. Several model categories are presented. And some open questions are discussed.

Thomas Leinster (Department of Mathematics, University of Glasgow) tl@maths.gla.ac.uk

Survey and Taxonomy

I will describe in as non-technical a way as possible some of the definitions of weak n-category that have been proposed, drawing attention to similarities rather than differences. The various definitions are placed in two groups, "algebraic" and "non-algebraic". Broadly, I will follow Chapter 10 of my book "Higher Operads, Higher Categories" (http://arxiv.org/abs/math.CT/0305049 or http://books.cambridge.org/0521532159.htm)

Thomas Leinster (Department of Mathematics, University of Glasgow) tl@maths.gla.ac.uk

Simplicial Definition
Slides: streettalk1.pdf    streettalk2.pdf

Joint with Nick Gurski.

Ross Street has proposed two related definitions of n-category, one in 1987 and one in 2002. Thomas Leinster will present the older definition; Nick Gurski will present the newer one and some of the motivating ideas.

Ross Street (Mathematics Department, Macquarie University) street@ics.mq.edu.au http://www.math.mq.edu.au/~street/

An Australian Conspectus of Higher Categories
Notes:
pdf

Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this talk, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higher-dimensional work.

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