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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 $A(\bd Y)$. |
395 (See \cite{1009.5025} and \cite{kw:tqft}; |
395 (See \cite{1009.5025} and references therein; |
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. |