Extended Topological Quantum Field Theory (ETQFT) by Jacob Lurie

Question

What is the functorial (categorical) definition of TQFT (Topological Quantum Field Theory), which Jacob Lurie "had extended", for his ETQFT ?

Actually I just need to know what are basic tools, to state properly a categorical definition of a classical TQFT. Thank you very much for your help !

Answer 1

This is just one definition of a TQFT. There are many more, but this is the one I personally have encountered before: Let $\mathbf{Bord}(n+1)$ be the category with objects closed, oriented, smooth $n$-manifolds and for $Y_1, Y_2$ two objects in this category, $Hom(Y_1,Y_2)$ be the set of $(n+1)$ bordism classes between $Y_1$ and $Y_2$. An $(n+1)$-TQFT is a symmetric monoidal functor from $\mathbf{Bord}(n+1)$ to the category $\mathbf{Vec}_{\mathbb{C}}$ of finite dimensional complex vector spaces.

More specifically: an $(n+1)$-TQFT is a pair $F=(V,Z)$ where $V(Y)$ assigns a finite dimensional complex vector space to a closed, oriented, smooth $n$-manifold $Y$ and $Z$ assigns to each $(n+1)$-bordism $X$ between two $Y_-,Y_+\in\mathbf{Bord}(n+1)$ a linear map $Z(X):V(Y_-)\to V(Y_+)$ such that:

1. If $X$ and $X'$ belong to the same bordism class, then $Z(X)=Z(X')$.
2. $Z(Y\times I)=id_{V(Y)}$
3. $Z(X_1\cup X_2)=Z(X_2)\circ Z(X_1):V(Y_1)\to V(Y_2)\to V(Y_3)$
4. $V(\emptyset)\cong\mathbb{C}$

5. $V(Y_1\sqcup Y_2)\cong V(Y_1)\otimes V(Y_2)$ with the following diagrams commuting: $$\require{AMScd} \begin{CD} V((Y_1\sqcup Y_2)\sqcup Y_3) @>{\cong}>> (V(Y_1)\otimes V(Y_2))\otimes V(Y_3);\\ @VVV @VVV \\ V(Y_1\sqcup( Y_2\sqcup Y_3)) @>{\cong}>> V(Y_1)\otimes(V(Y_2)\otimes V(Y_3)); \end{CD}$$

   $$\require{AMScd} \begin{CD} V(\emptyset\sqcup Y) @>{\cong}>> \mathbb{C}\otimes V(Y);\\ @VVV @VVV \\ V(Y) @>{=}>> V(Y); \end{CD}$$

6. There is an isomorphism $V(Y_1\sqcup Y_2)\cong V(Y_2\sqcup Y_1)$ such that the following diagram commutes: $$\require{AMScd} \begin{CD} V(Y_1\sqcup Y_2) @>{\cong}>> V(Y_1)\otimes V(Y_2);\\ @VVV @VVV \\ V(Y_2\sqcup Y_1) @>{\cong}>> V(Y_2)\otimes V(Y_1); \end{CD}$$

Subject to these axioms, the functor $F=(V,Z)$ can be anything we like; what we take it to be determines the TQFT we are looking at. There is a theorem which says that: $Z(M)$ is a smooth invariant for closed $(n+1)$-manifolds $M$ and $V(Y)$ is a representation of the mapping class group of $Y$. As far as I am aware, given a TQFT $F=(V,Z)$, it is still an open (and very difficult) question to determine exactly how $F$ relates to 'classical' invariants such as, for example, homology groups or homotopy groups.