A Fields Institute Sponsored Workshop
Smooth Structures in
Logic, Category Theory and Physics
of Mathematics & Statistics
May 1-3, 2009
Note: Some Slides of Talks
Since the 1970's, studies of
categories of "smooth spaces", i.e. spaces with sufficient
structure to define
been developed by numerous researchers. Such settings make possible,
for example, a more general,
abstract version of differential geometry. One of the issues that the
more categorically minded
researchers have investigated is the fact that the category of
manifolds and smooth maps
is not cartesian closed, i.e. the collection of smooth functions
between two manifolds is not itself a manifold. Many of these approaches overcome this,
and thus the smooth maps between smooth spaces form a smooth space.
The work of A. Frölicher, A.
Kriegl, and P. Michor generalizes calculus in Banach spaces to convenient vector
spaces, a class of vector
spaces equipped with enough abstract structure to allow a definition of
smooth maps. The category of convenient vector spaces and smooth maps
is cartesian closed. Moreover, such a space may be viewed
simultaneously as a bornological space, a locally convex space, and as
an ell-infinity space, all of which
have been proposed as possible settings to study smoothness. Similarly, following
early work of K. Chen, J.-M. Souriau, and others, categories of
smooth spaces containing the
category of C-infinity
manifolds have been considered as suitable abstract settings for doing
The development of synthetic
differential geometry and smooth infinitesimal
analysis by F.W. Lawvere, A.
Kock, I. Moerdijk, G.
Reyes, and others is based on a development of a categorical theory of
infinitesimals in appropriate categories of spaces (smooth toposes).
Such categories contain the category of C-infinity manifolds and
In a different direction, recently
T. Ehrhard and L. Regnier developed the differential
lambda-calculus motivated by linear logic, differential calculus,
and Ehrhard's work on various locally convex topological models of
linear logic. This
inspired the recent development of differential
categories by R. Blute, R.
Cockett, and R. Seely.
Finally, recent work by J.
Baez and U. Schreiber in theoretical physics exploits these more
abstract versions of differential geometry to avoid the technical
difficulties implicit in the theory of infinite-dimensional manifolds.
This leads to their notion of higher gauge theory.
In this workshop, we propose to
bring experts together from these different areas to encourage further
interaction in the study of smooth structures in categories,
logic. Several of our invited speakers will be asked to give tutorials
in order to make the subject accessible to students and other new
researchers. In light of the recent interest in these topics
(especially in the category theory community) there is little doubt
that the workshop will act as a stimulus for further discussion and
* John Baez (UC Riverside)
* Richard Blute (Ottawa) (Tutorial)
* Kristine Bauer (UCalgary)
Robin Cockett (Calgary)
* Thomas Ehrhard (PPS Paris)
* Alexander Hoffnung
* Anders Kock
* Dorette Pronk (Dalhousie)
* Gonzalo Reyes
* Konrad Waldorf
* Andrew Stacey
Abstracts of Talks <--- New: Some Slides of
is intended to be
workshop, with student participation in mind, including introductory
tutorials, with plenty of time to talk with the speakers. (Please
register on the Fields Webpage).
Anyone interested in attending is requested to contact one
organizers as soon as possible.
For official preconference registration,
Fields Webpage for Smooth Structures Workshop
613-562-5800, ext. 3535 (firstname.lastname@example.org)