# Examples of higher dimensional TQFTs

1-dimensional TQFT's assign to every 1-manifold (disjoint union of circles) a vector space and to every surface a linear map between the vector spaces that correspond to the boundary manifolds. So this is a functor from the cobordism category of 1-manifolds to the category of vector spaces. What are some examples of higher dimensional TQFT's? ie functors from the cobordism category of n-manifolds to the category of vector spaces.

• As already said, two dim TQFT corresponds to Frobenius algebra, because every surface has a "pair of pants" decomposition. The simplest example of $2$ dim TQFT I know : take a triangulated surface, a vector space $V$, an element $a \in V \otimes V \otimes V$ and a contraction $\mu \in V^* \otimes V^*$. Triangulating your surface, you can put a copy of $a$ on every triangle, taking the big tensor product of all $a$ and using edges for contracting. You will end with a scalar which is an invariant of the surface. $a$ and $\mu$ have to verify some conditions (invariance under Pachner moves). – user171326 Jun 29 '17 at 17:12

Extended TQFTs are a technically more daunting, but conceptually easier to define gadget: if you believe this paper (you should; a more user-friendly presentation is contained in this master thesis) the $\infty$-category (in some model: the paper uses Segal spaces -am I right?-) of $n$-dimensional cobordisms happen to be the free symmetric monoidal $(\infty,n)$-category on a single generator.