(Higher) Categories for the working physicist

Rafał R. Suszek


Construction of field theories in the point like—particle paradigm uses structures 
such as a fibre bundle with connection, a (Lie) group, a (Lie) algebra, a (pre)symplectic 
manifold, and — in the abstract (and therefore typically difficult to realise explicitly) 
approach to quantisation — a certain covariant functor from the geometric category that 
models particle(-field) propagation in the ambient spacetime to the algebraic category 
of Hilbert spaces (usually with some additional structure). Thus, standard field theory 
requires at most the concept of a category. Studies of the (hypothetical) dynamics 
of extended objects endowed with a topological charge (loops, membranes etc.), 
on the other hand, seem to call, and naturally at that, for a categorification resp. 
internalisation of essentially all of these concepts, whence a wealth of (typically weak) 
higher categories in any rigorous discussion of explicit models, such as lagrangean 
(super)string theory, topological field theory (e.g., Chern—Simons theory) etc.

In my talk, I shall review some fundamental instances of this universal phenomenon 
of categorification, staying mainly in the best-understood environment of two-dimensional
field theory (and so in the bicategorial context), with — time permitting — minor 
detours into the realm of higher-dimensional constructs.