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Macquarie University  Department of Mathematics

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Workshop on Categorical Methods in Algebra, Geometry and Mathematical Physics

Satellite to the StreetFest conference in honour of Ross Street's sixtieth birthday

July 18-21 2005, Australian National University, Canberra

Tue, 19 July:  11:30 - 12:10

Differential and Smooth Categories
Cockett, Robin (University of Calgary)

\thanks{This is joint work with Richard Blute and Robert Seely, both
of whom join me in wishing Ross Happy Birthday!}
Recently T. Ehrhard and L. Regnier introduced
the ``differential $\lambda$-calculus" as an abstract
syntax for differentiation. Ehrhard also introduced
semantic settings which modeled the calculus
(e.g. Finiteness and Koethe spaces).

This talk will address the categorical semantics of
differentiation in the spirit of the above. However,
we shall take the opportunity to generalize those ideas
so that standard models of differentiable functions
(from first year calculus) are included. One effect
of this generalization is to remove the necessity for
higher order constructs (often absent from simpler
models): so it seems appropriate to miss out the
$\lambda$ and say that these categories model the
``differential calculus".

A differential category is an additive category with a
coalgebra modality and a differential combinator.
A coalgebra modality is a comonad in which each cofree
coalgebra of the modality is a comonoid. The differentiable
or smooth functions are the maps in the
coKleisli category. The coKleisli category of a
differentiable category is a smooth category. We
shall show how, under suitable conditions, one can
recover an underlying differential category from
a smooth category.

Typeset PDF of this abstract.