TQFT which extends down to points we use the language of higher category theory; see [@L] for an introduction and for much more exposition about the $n$-categories we now introduce. Let ${\mathbf{Bord}^{SO}}_n$ denote the $n$-category[^10] whose objects are finite unions of oriented points, 1-morphisms are oriented bordisms of objects, 2-morphisms are oriented bordisms of 1-morphisms, and so forth.[^11] Note ${\mathbf{Bord}^{SO}}_n$ carries a symmetric monoidal structure given by disjoint union. Let ${\mathcal{C}}$ be any symmetric monoidal $n$-category.

\[thm:23\] An *extended $n$-dimensional TQFT with values in ${\mathcal{C}}$* is a symmetric monoidal functor $F{\colon}{\mathbf{Bord}^{SO}}_n\to{\mathcal{C}}$.

The 0-1-2 finite gauge theory can be constructed by hand.

0-1-2 theories for general $G$ {#subsec:2.2}
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Let $F$ be a fully extended topological field theory of dimension $n$. We can regard $F$ as a prescription for assigning invariants to manifolds with corners of all dimensions $\leq n$, together with set of rules for how these invariants behave when we glue manifolds together. Since every smooth manifold $M$ can be assembled by gluing together very simple pieces (for example, by choosing a triangulation of $M$), we might imagine that the value of $F$ on arbitrary manifolds is determined by its values on a very small class of manifolds. In 