Abstrakt:
Atiyah and Segal's axiomatic approach to topological and conformal quantum field theories provided a beautiful link between the geometry of "spacetimes" (cobordisms) and algebraic structures. Combining this with the physical notion of "locality" led to the introduction of the language of higher categories into the field. I will give an introduction to the topic and explain a natural generalization thereof, called "twisted" field theories by Stolz-Teichner. The definition uses the notion of lax or oplax natural transformations of strong functors of higher categories for which I will sketch a framework. I will discuss the fully extended case, which gives a comparison to to Freed-Teleman's "relative" boundary field theories. Finally, I will explain some simple examples, one of which comes from units of (E_n-)algebras and the other from endomorphisms of a vector space (and generalizations). This is joint work with Calaque, Gwilliam, and Johnson-Freyd.