Atiyah axiomatized a TQFT as a (symmetric monoidal) functor from a certain cobordism category to the category of vector spaces. This can be described explicitly in dimension 2, because one can decompose any surface into pants, which are manifolds with boundary but without corners.

In higher dimensions, it is more convenient to allow the decomposition pieces to have corners, so that we can perform triangulations. Then the cobordism category has to be described as an ($\infty,n$)-category. Allowing gluing along triangulation guarantees that a TQFT is fully local, and depends only on its value at a point.

If we replace the target category of vector spaces by the categpry of $E_n$-algebras, then the TQFT can be described explicitly by factorization homology, which is a homotopy integration process.

The level of technicality of this talk is given by the function $f(t) = \tan \frac{(\pi-\epsilon)t^2}{2}$, $t \in [0,1]$, so hopefully everyone is addressed.