A spherical finite tensor category with a modified trace gives a finite-dimensional open-closed 3d TFT via a new handlebody invariant and the universal construction.
On piecewise linear cell decompositions
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abstract
In this note, we introduce a class of cell decompositions of PL manifolds and polyhedra which are more general than triangulations yet not as general as CW complexes; we propose calling them PLCW complexes. The main result is an analog of Alexander's theorem: any two PLCW decompositions of the same polyhedron can be obtained from each other by a sequence of certain "elementary" moves. This definition is motivated by the needs of Topological Quantum Field Theory, especially extended theories as defined by Lurie.
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Non-semisimple open-closed 3d TFT
A spherical finite tensor category with a modified trace gives a finite-dimensional open-closed 3d TFT via a new handlebody invariant and the universal construction.