Joao Faria Martins (University of Leeds) – A once-extended TQFT categorifying Quinn's finite total homotopy TQFT

Quinn's Finite Total Homotopy TQFT is a TQFT (topological quantum field theory) defined for any dimension, $n$, of space, and that depends on the choice of a homotopy finite space, $B$. For instance, $B$ can be the classifying space of a finite group or of a finite 2-group.
I will report on recent joint work with Tim Porter on once-extended versions of Quinn's Finite total homotopy TQFT, taking values in the (symmetric monoidal) bicategory of groupoids, linear profunctors, and natural transformations between linear profunctors. The construction works in all dimensions, thus in particular it yields (0,1,2)-, (1,2,3)- and (2,3,4)-extended TQFTs, any time we are given a homotopy finite space $B$. I will show how to compute these once-extended TQFTs for the case when $B$ is the classifying space of a homotopy 2-type, represented by a crossed module of groups.

References: Faria Martins J, Porter T : A categorification of Quinn's finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from strict omega-groupoids. arXiv:2301.02491 [math.CT].