## Hyperconnections

### Robin Cockett – 20 March 2019

The purpose is this time is to explain the observations Richard Garner and I have been stumbling into during my visit. In particular, to explain the Galois adjunction between join restriction categories over a manifold complete category, B, and internal partite categories (and cofunctors) within B. The adjunction specializes to an isomorphism between hyperconnections over B (that is hyperconnected join functors to B) and hyperconnections within B (that is source etale partite categories within B).

The observation seems to unify a number of observations in the literature ranging through semigroup theory, algebraic geometry, to differential topology. Hopefully, I will mention some of these!

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