The purpose of the talk is to lay (some of) the groundwork for observations Richard Garner and I have been stumbling into during my visit.
In particular, we observed that there is a Galois adjunction between join restriction categories (and functors) over a manifold complete category, B, and internal partite categories (and cofunctors) within B. The adjunction specializes to an isomorphism between join restriction categories hyperconnected over B and source etale partite categories within B. The observation seems to unify a number of similar observations in the literature ranging from semigroup theory to differential topology.
The story behind these ideas started about 30 years ago when Marco Grandis (using rather different nomenclature) introduced the manifold construction and manifold complete categories. The first talk will be aimed at developing some of this "abstract topology" for join restriction categories in so far as it is required to support the main observations above.