  We investigate the relationship between the algebra of tensor categories and
the topology of framed 3-manifolds. On the one hand, tensor categories with
certain algebraic properties determine topological invariants. We prove that
fusion categories of nonzero global dimension are 3-dualizable, and therefore
provide 3-dimensional 3-framed local field theories. We also show that all
finite tensor categories are 2-dualizable, and yield categorified 2-dimensional
3-framed local field theories. On the other hand, topological properties of
3-framed manifolds determine algebraic equations among functors of tensor
categories. We show that the 1-dimensional loop bordism, which exhibits a
single full rotation, acts as the double dual autofunctor of a tensor category.
We prove that the 2-dimensional belt-trick bordism, which unravels a double
rotation, operates on any finite tensor category, and therefore supplies a
trivialization of the quadruple dual. This approach produces a quadruple-dual
theorem for suitably dualizable objects in any symmetric monoidal 3-category.
There is furthermore a correspondence between algebraic structures on tensor
categories and homotopy fixed point structures, which in turn provide
structured field theories; we describe the expected connection between pivotal
tensor categories and combed fixed point structures, and between spherical
tensor categories and oriented fixed point structures.
