Abstracts for Higher Geometric Structures along the Lower Rhine

A Segal space is a simplicial topological space $X = (X_n)$ with a certain condition expressing $X_n$ as an $n$-fold homotopy fiber product of $X_1$ over $X_0$. This concept can be seen as encoding a weak (higher) categorical structure.

I will present some basic notions and results of "global" homotopy theory, where objects are spectra (in the sense of stable homotopy theory) equipped with actions of all compact Lie groups, in a way compatible with transfers and restriction along group homomorphisms.

The precise implementation proceeds via a new model structure on the category of orthogonal spectra, with "global equivalences" as weak equivalences.

Toposes and C$^*$-algebras both represent generalized concepts of space. The utmost generality seems achieved by studying C$^*$-algebras in (Grothendieck) toposes, as first done by Banaschewski and Mulvey. In the commutative case, Gelfand duality holds and one gets interesting examples of internal locales.