390 For our purposes, an $n{+}1$-dimensional TQFT is a locally defined system of

   390 For our purposes, an $n{+}1$-dimensional TQFT is a locally defined system of

   391 invariants of manifolds of dimensions 0 through $n{+}1$. In particular,

   391 invariants of manifolds of dimensions 0 through $n{+}1$. In particular,

   392 the TQFT invariant $A(Y)$ of a closed $k$-manifold $Y$ is a linear $(n{-}k)$-category.

   392 the TQFT invariant $A(Y)$ of a closed $k$-manifold $Y$ is a linear $(n{-}k)$-category.

   393 If $Y$ has boundary then $A(Y)$ is a collection of $(n{-}k)$-categories which afford

   393 If $Y$ has boundary then $A(Y)$ is a collection of $(n{-}k)$-categories which afford

   394 a representation of the $(n{-}k{+}1)$-category $A(\bd Y)$.

   394 a representation of the $(n{-}k{+}1)$-category $A(\bd Y)$.

   396 for a more homotopy-theoretic point of view see \cite{0905.0465}.)

   396 for a more homotopy-theoretic point of view see \cite{0905.0465}.)

   397 

   397 

   398 We now comment on some particular values of $k$ above.

   398 We now comment on some particular values of $k$ above.

   399 A linear 0-category is a vector space, and a representation

   399 A linear 0-category is a vector space, and a representation

   400 of a vector space is an element of the dual space.

   400 of a vector space is an element of the dual space.