M. Atiyah defined an \(n\)-dimensional Topological Field Theory (TFT) as a functor \(\mathsf{F}\) from a certain geometrical defined category \(\text{Bord}_n\) to the category of complex vector spaces. The objects of \(\text{Bord}_n\) are compact oriented \((n-1)\)-manifolds and the morphisms are oriented bordisms. The functor \(\mathsf{F}\) assigns a vector space to any \((n-1)\)-manifold and a linear map to a bordism.
An Extended TFT is obtained by extending a TFT functor to lower dimensional manifolds. It is reasonable to require that the functor \(\mathsf{F}\) assigns a \(\mathbb{C}\)-linear category to an \((n-2)\)-manifold, a \(\mathbb{C}\)-linear 2-category to an \((n-3)\)-manifold, etc. The obvious reason that an axiomatic definition of an Extended TFT has not been formulated is the lack of a universally accepted definition of an \(n\)-category for \(n>2\) which is due to the lack of understanding the physical meaning of higher categories.
The paper under review goes back to the physical roots of higher categories. First it discusses 2-dimensional TFTs and explains why boundary conditions in a 2d TFT form a category. Second it explains why boundary conditions in a 3d TFT form a 2-category. Then it explains why an \(n\)-dimensional TFT functor assigns a \(\mathbb{C}\)-linear \((k-1)\)-category to a compact oriented \((n-k)\)-manifold from a more physical viewpoint. Two examples of TFTs are discussed at last: the Rozansky-Witten model as a 3d TFT and its application in the classification of monoidal deformations of the derived category of coherent sheaves and the 4d Topological Gauge Theory and its application in the Geometric Langlands Program.
For the entire collection see [Zbl 1220.00033].

Reviewer: Ren Guo (Corvallis)