Abstracts:  Fields Smooth Structures Workshop
                                                Ottawa (May 1-3, 2009)

                 Some Slides of Talks are Available Below (Authors Names in Blue)

John Baez:  Why Smooth Spaces?

The category of smooth manifolds and smooth maps is often taken as the default context for research on differential geometry.  However, in many applications it is convenient or even necessary to work in a more general framework. In this introductory talk we explain various reasons for this. We also describe the advantages of various alternative frameworks, including Banach manifolds and other types of infinite-dimensional manifolds, Chen spaces and diffeological spaces, orbifolds and differentiable stacks, and synthetic differential geometry.

Kristine Bauer:  Introduction to functor calculus

This talk aims to give an introduction to the Goodwillie-Weiss calculus of functors.   The calculus of functors can be applied in several different settings, including homotopy calculus, which deals with functors whose domain and target categories are either topological spaces or spectra, or manifold calculus which deals with functors whose domain is the poset of open subsets of a fixed manifold M and whose target is topological spaces.   The calculus of functors provides "polynomial" approximations to a functor.  In particular,  we can linearize a functor in the same manner that differential calculus provides a linear approximation for functions.  For example, in the homotopy case a reduced (sends a point to a point) functor which preserves weak equivalences is linear if it is excisive; i.e. it takes homotopy pushout squares to homotopy pullback squares.  Linear functors are particularly nicely behaved with respect to the suspension operation, and hence linearization is the same as stabilization in the homotopy case.  An appropriate notion of linear is similarly defined in the other cases.

This expository introduction to the calculus of functors will begin by introducing excisive and n-excisive functors, and by explaining how to build a "Taylor tower" of  n-excisive functors which approximate the original functor, just as the Taylor polynomials approximate a function.   We will present the general theory whenever possible, but when specialization is required, we give special attention to manifold calculus.   In the second part (time permitting), we will survey some of the applications of calculus, in particular manifold calculus.


Rick Blute:   From Linear Logic To Differential Categories

We give an introductory exposition of Girard's linear logic and its categorical semantics. This will be preliminary to introducing the theory
of differential categories. Differential categories, due to Blute, Cockett and Seely, were a categorical axiomatization of differential linear logic
and differential lambda-calculus, due to Ehrhard and Regnier.

We will give the basic definitions and some examples, and introduce the more recent theory of Kahler categories, which introduces differentiation
via a universal property, similar to Kahler differentiation in algebraic geometry.

Robin Cockett:   Introduction to Cartesian Differential Categories   Robin Cockett's Slides

Somewhere at the bottom of the food chain in the world of smoothness are Cartesian Differential Categories (CDC): a basic quite fundamental structure.   The paradigmatic example of a CDC is finite dimensional real vector spaces with smooth (infinitely differentiable) maps.   Notably the coKleisli category of any  (commuting) Differential Category is a CDC.  There is a term logic for CDCs which will instantly be familiar to anyone who has worked with multidimensional calculus.  The term logic allows a description of the free differential categories and facilitates the description of standard (e.g. bundle) constructions on differential categories.   (Joint work with Rick Blute and Robert Seely)


Thomas Ehrhard:  Differential Linear Logic and Computation     

Starting from a model of linear logic based on a particular class of linearly topologized vector spaces, we introduce differential linear logic - an extension of ordinary linear logic by new rules on the exponentials - and two related formalisms: the resource lambda-calculus and differential interaction nets. We show how these formalisms give a new viewpoint on abstract machines (for the lambda-calculus) and concurrency.

Alexander Hoffnung:    From Smooth Spaces to Smooth Categories     Alexander Hoffnung's Slides

Various generalizations of smooth manifolds and smooth maps have nice categorical properties. In this talk, we will detail some of these properties in the case of diffeological spaces, while seeing that the usual category of smooth manifolds sits nicely in this framework. We can define constructions one expects from differential geometry such as vector fields and differential forms on diffeological spaces, with all the familiar operations. Further, we see that using smooth spaces we can define a notion of smooth category. If time permits, we will describe an application of smooth categories to principal G-bundles.

Anders Kock: Part 1: Kaehler differentials for Fermat theories   

A Fermat theory is an algebraic theory T (in the sense of Lawvere) with a certain property, making  "partial differentiation" an intrinsic (canonical)
structure in it. Examples are the theory of commutative rings, and also the theory of smooth functions.  There is a notion of a module over a
Fermat algebra, and of a derivation into such. For a given algebra A, there is a universal module Omega_A  receiving a derivation from it. This is the module of Kaehler differentials. We describe in particular the Kaehler differentials of a free algebra S(V), which turns out to be the tensor product  S(V)@V.

Anders Kock:  Part 2:  Synthetic meaning of Kaehler differentials.

We describe the context of synthetic differential geometry, and some particularly simple topos models for it, built out of  Fermat theories. In this
context, the objects ("spaces") M carry a reflexive symmetrive "neighbour"-relation, in terms of which the notion of  differential form can be described in simple combinatorial terms. There exists in such a topos an object Omega such that differential 1-forms on  M are in bijective correspondence with maps 
M --> Omega. A certain  subspace of Omega similarly classifies closed differential 1-forms, and is therefore a kind of Eilenberg-Mac Lane space K(R,1).

(The above work of Anders Kock is partly joint with E. Dubuc)

Dorette Pronk:  Equivariant Homotopy Theory for Smooth Orbifolds

Smooth orbifolds are paracompact spaces which can locally be described as the quotient of an open subset of Euclidean space by the smooth action of a finite group. An orbifold is called representable if it can be described as the quotient of a manifold by the action of Lie group (in this case, the group does not need to be finite, but the isotropy groups are required to be finite.) Since representable orbifolds have the same underlying geometric structure as smooth G-spaces we would like to extend equivariant homotopy theory to obtain new orbifold homotopy invariants. In general, maps between orbifolds are more general than maps between G-spaces. So we need to conditions on equivariant invariants to be orbifold invariants.

In order to understand what maps between orbifolds should be, we represent orbifolds by Lie groupoids. These Lie groupoid representations for a given orbifold are only unique up to Morita equivalence.  An orbifold is representable if it can be represented by a (non-unique) translation Lie groupoid. So in order for us to be able to import equivariant homotopy invariants into orbifold homotopy theory, we need to know how maps between translation Lie groupoids are related to equivariant maps and we need a manageable description of what it means to be invariant under Morita equivalence.

In this talk I will show that the 2-category of smooth representable orbifolds can be described as a bicategory of fractions of the 2-category of translation groupoids and equivariant maps, and I will give a set of generators for the Morita equivalences which is easy to work with. As an application, I will show how one can define orbifold Bredon cohomology and (geometric) orbifold K-theory.

This is joint work with Laura Scull from Fort Lewis College in Durango, Colorado.

Gonzalo Reyes:  Analysis in Smooth Toposes: Knowledge and Conjecture

Andrew Stacey:  Comparative Smootheology     Andrew Stacey's Slides

Smooth manifolds are extremely nice spaces. The fact that they have charts means that a vast amount of the theory of Euclidean spaces can be easily transferred to manifolds. This makes for a very useful subject.

However, the charts also make manifolds very fragile: it is easy to do something to a manifold that makes it no longer a manifold. Taking a quotient by a group action is one such, looking at mapping spaces is another. Often, specific operations can be fixed - orbifolds fix the quotienting, infinite dimensional manifolds fix the mapping spaces - but systematic case-by-case fixing is a little unsatisfying. Over the years there have been several attempts to build a suitable category of "smooth objects" generalising smooth manifolds. The general method is to take some property that all manifolds have, which can be defined in a more robust way than charts.

In this talk I shall review some of these attempts, focussing particularly on the similarities between them. I shall try to motivate my own favourite: Frölicher spaces. In addition, it is worth mentioning that the majority of these categories come under the heading of "sets with structure". There have also been attempts to do away with the "sets with" part of this and I shall talk about why one might wish to do this.

This is based partly on arxiv:0802.2225 [math.DG].

Konrad Waldorf:  Smooth Functors for higher-dimensional Parallel Transport

A principal bundle with connection can entirely be described by its parallel transport, characterized as a locally smooth functor. I will talk about notions of locality and smoothness that can systematically be categorified, based on work of Baez/Schreiber and joint work with Urs Schreiber. This leads to a new way to understand parallel transport along surfaces and connections on possibly non-abelian gerbes.