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For a topological quantum field theory, $Z:Cob(n)\to Vect(\mathbb{C})$ why is it that typically $Z(\emptyset)\cong \mathbb{C}$? Is that just the definition that makes everything work?

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Look at disjoint union of a manifold and an empty manifold. So you're getting $V$ and $C$ so when you combine you better get $V$ again. This is the monoidal unit requirement.

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