This lecture course will provide an introduction to the topos approach to quantum theory and, more generally, to the formulation of physical theories. The topos approach was initiated by Isham and Butterfield in the late 1990s and was substantially developed over the last few years by Isham, me, and a number of other researchers. The approach does not just draw on category and topos theory, but also uses operator algebras, logic, and of course foundations of physics in a fundamental way. My aim is to present all these aspects in a way that is accessible to an interested master level student.

The structure of the lecture course will be as follows:

1 Motivation

2 Foundations of quantum theory and foundations of physics (propositions, states, quantum logic, hidden variable theories, contextuality and Kochen-Specker theorem, locality and Bell's theorem, probabilites and truth values, instrumentalism vs. realism)

3 Basics of category and topos theory (definitions, limits and colimits, universal constructions, adjunctions, Galois connections, presheaves and sheaves, topoi, subobject classifier and internal logic, examples)

4 Some aspects of the theory of C*-algebras and von Neumann algebras (definitions, representations, states, examples, types of von Neumann algebras, abelian algebras and Gelfand duality, Jordan algebra and Lie algebra structures, orientations, physical interpretation)

5 Basic structures in the topos approach (contexts, spectral presheaf, clopen subobjects, Heyting and bi-Heyting algebras, pseudostates, truth value assignment, states and probability measures)

6 Advanced structures and results in the topos approach (internalisation of probabilities, time evolution, generalised Gelfand duality and noncommutative geometry)