theory. Let $({\textnormal{Bord}^{SO}}_n,\amalg)$ be the bordism category whose morphisms are oriented $n$-manifolds; it carries a symmetric monoidal structure given by disjoint union. Let $({\operatorname{Vect}{^{\vphantom{1*\prime y}}}_{{{\mathbb C}}}},\otimes )$ denote the symmetric monoidal category of finite dimensional complex vector spaces under tensor product.

\[thm:2\] An *$n$-dimensional topological quantum field theory* $F$ on oriented manifolds is a symmetric monoidal functor $$F{\colon}({\textnormal{Bord}^{SO}}_n,\amalg )\longrightarrow ({\operatorname{Vect}{^{\vphantom{1*\prime y}}}_{{{\mathbb C}}}},\otimes ).$$

Thus a field theory assigns to each closed oriented $(n-1)$-manifold $Y$ a complex vector space $F(Y)$ and to each closed oriented $n$-manifold $X$ a number $F(X)\in {{\mathbb C}}$. If $X{\colon}Y_0\to Y_1$ is a compact manifold with boundary[^8] $-Y_0\amalg Y_1$, then $F(X){\colon}F(Y_0)\to F(Y_1)$ is a linear map. The functor $F$ maps the gluing operation of bordisms to the composition of linear maps. We refer the reader to [@A], [@S], [@L] for more details and exposition about this definition.

There are many possible variations on Definition \[thm:2\]. For the domain we may take a bordism category of manifolds with other topological structures: oriented manifolds, spin manifolds, framed manifolds, etc. There are also other choices for the codomain. This definition describes “two-tiered theories”; we will see more tiers in §\[sec:2\]. If, say, $n=3$, then we designate this a ‘2-3 theory’ 