Categorified and manifestly homotopy-invariant TQFTs

TQFTs can be understood as numerical invariants of smooth manifolds (with possibly extra structure) that can be computed in terms of gluing a manifold from simpler pieces. The invariants that TQFTs usually produce are numerical only. So the question that we try to address is, is it possible to categorify these numerical invariants to algebraic ones, depending functorially on its argument?

Besides providing possibly finer invariants, this will also have another advantage: any TQFT that is ‘categorified’ in this sense is manifestly homotopy-invariant essentially by definition.

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