Homotopy et TQFTs

 The essential idea of TQFTs is that they are a symmetric monoidal functor $\mathcal{Z}$ from the category of topological spaces, here $n$-bordisms (with a number of technicalities suppressed for now):

\[\mathcal{Z}\;:\; \text{Bord}_{n}\;\longrightarrow \;\text{Vect}_{\mathbb{K}}\;.\]

$\mathcal{Z}$ is functorial w.r.t orientation preserving diffeomorphisms of $\Sigma $, an oriented smooth $D$-manifold and $M$, a $D+1$ manifold. The general approach is by defining a homotopy axiom and an additive axiom, in the sense that one can attribute the above functorial definition. In the TQFTs the homotopy axiom, which has to do with cylinders, is replaced by cobordisms instead -- established by Atiyah in his paper on topological quantum field theories. One can now use this as a starting point and define homotopy quantum field theories (HQFTs) as the following alteration of the above definition: taking $\mathbf{B}$-cobordisms, one can define the symmetric monoidal category $\textbf{Hcobord}(n, \mathbf{B})$. Then, an HQFT is a functor:

\[\mathcal{Z}^{\mathcal{H}}\;:\;\textbf{Hcobord}(n, \textbf{B})\; \longrightarrow \;\text{Vect}_{\mathbb{K}}\;.\]

There are some more aspects about $\text{Vect}_{\mathbb{K}}$ that are of importance, but I do not have an understanding strong enough to explain them.