[Alta-Logic] Peripatetic seminar: Tangent structure and synthetic differential geometry

[Robin Cockett]

Time: 18th April, Wednesday 3:00pm

Place: ICT 616 (University of Calgary)

Speaker: Robin Cockett

Title: Tangent structure and synthetic differential geometry

Abstract:

I am going to describe some aspects of current work with Geoff
Cruttwell on tangent structure.

A category with tangent structure is an abstract setting for
differential geometry.  The notion was first proposed by Jiri Rosicky
in 1984 in a paper remarkable for its brevity.  Rosicky promised a
follow on paper which would contain details of proofs, however, it
seems that this never happened.   Geoff and I stumbled into this area
as we were looking for abstract characterizations for categories of
differential manifolds. Anders Kock kindly pointed out Rosicky's
remarkable paper to us.

A particularly, fascinating aspect of Rosiky's paper is that he
initiates the attempt to completely characterize synthetic
differential geometry (SDG) as "representable" tangent structure.  As
we had not even seriously considered the possibility that these SDG
settings would be included under this sort of abstract formulation,
Geoff and I have been scrambling to understand this connection.
Approaching SDG though representable tangent structure provides a
rather different perspective as to what these models bring to the
table and this --together with (I hope) a discussion of a particular
model -- will be the main focus of the talk.

 What seems quite remarkable is that the concept of tangent structure
seems to subsumes ALL the notions of differential geometry of which we
are aware!