Topos theory and C$^*$-algebras

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. As a case in point, any unital C$^*$-algebra $A$ (in Sets) defines a commutative C$^*$-algebra in the topos of sheaves on the poset of commutative unital C$^*$-subalgebras of $A$, whose Gelfand spectrum can be computed explicitly (joint work with Martijn Caspers, Chris Heunen, Bas Spitters, and Sander Wolters). This interaction between topos theory and C$^*$-algebras also leads to new questions about the latter with potential relevance to the former (notably new examples of sites).