165 

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

   166 

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

   167 \dropcap{T}opological quantum field theories (TQFTs) provide local invariants of manifolds, which are determined by the algebraic data of a higher category.

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

   168 

   168 (See \cite{1009.5025} and \cite{kw:tqft};

   169 An $n+1$-dimensional TQFT $\cA$ associates a vector space $\cA(M)$

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

   170 (or more generally, some object in a specified symmetric monoidal category)

   170 

   171 to each $n$-dimensional manifold $M$, and a linear map

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

   172 $\cA(W): \cA(M_0) \to \cA(M_1)$ to each $n+1$-dimensional manifold $W$

   172 By convention, a linear 0-category is a vector space, and a representation

   173 with incoming boundary $M_0$ and outgoing boundary $M_1$.

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

   174 An $n+\epsilon$-dimensional TQFT provides slightly less;

   174 So a TQFT assigns to each closed $n$-manifold $Y$ a vector space $A(Y)$,

   175 it only assigns linear maps to mapping cylinders.

   175 and to each $(n{+}1)$-manifold $W$ an element of $A(\bd W)^*$.

   176 

   176 In fact we will be mainly be interested in so-called $(n{+}\epsilon)$-dimensional

   177 There is a standard formalism for constructing an $n+\epsilon$-dimensional

   177 TQFTs which have nothing to say about $(n{+}1)$-manifolds.

   178 TQFT from any $n$-category with sufficiently strong duality,

   178 For the remainder of this paper we assume this case.

   179 and with a further finiteness condition this TQFT is in fact $n+1$-dimensional.

   179 

   180 \nn{not so standard, err}

   180 When $k=n-1$ we have a linear 1-category $A(S)$ for each $(n{-}1)$-manifold $S$,

   181 

   181 and a representation of $A(\bd Y)$ for each $n$-manifold $Y$.

   182 These invariants are local in the following sense.

   182 The gluing rule for the TQFT in dimension $n$ states that

   183 The vector space $\cA(Y \times I)$, for $Y$ an $n-1$-manifold,

   183 $A(Y_1\cup_S Y_2) \cong A(Y_1) \ot_{A(S)} A(Y_2)$,

   184 naturally has the structure of a category, with composition given by the gluing map

   184 where $Y_1$ and $Y_2$ and $n$-manifolds with common boundary $S$.

   185 $I \sqcup I \to I$. Moreover, the vector space $\cA(Y \times I^k)$,

   185 

   186 for $Y$ and $n-k$-manifold, has the structure of a $k$-category.

   186 When $k=0$ we have an $n$-category $A(pt)$.

   187 The original $n$-category can be recovered as $\cA(I^n)$.

   187 This can be thought of as the local part of the TQFT, and the full TQFT can be constructed of $A(pt)$

   188 For the rest of the paragraph, we implicitly drop the factors of $I$.

   188 via colimits (see below).

   189 (So for example the original $n$-category is associated to the point.)

   189 

   190 If $Y$ contains $Z$ as a codimension $0$ submanifold of its boundary,

   190 We call a TQFT semisimple if $A(S)$ is a semisimple 1-category for all $(n{-}1)$-manifolds $S$

   191 then $\cA(Y)$ is natually a module over $\cA(Z)$. For any $k$-manifold

   191 and $A(Y)$ is a finite-dimensional vector space for all $n$-manifolds $Y$.

   192 $Y = Y_1 \cup_Z Y_2$, where $Z$ is a $k-1$-manifold, the category

   192 Examples of semisimple TQFTS include Witten-Reshetikhin-Turaev theories, 

   193 $\cA(Y)$ can be calculated via a gluing formula,

   193 Turaev-Viro theories, and Dijkgraaf-Witten theories.

   194 $$\cA(Y) = \cA(Y_1) \Tensor_{\cA(Z)} \cA(Y_2).$$

   194 These can all be given satisfactory accounts in the framework outlined above.

   195 

   195 (The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories in order to be

   196 In fact, recent work of Lurie on the `cobordism hypothesis' \cite{0905.0465}

   196 extended all the way down to 0 dimensions.)

   197 shows that all invariants of $n$-manifolds satisfying a certain related locality property

   197 

   198 are in a sense TQFT invariants, and in particular determined by

   198 For other TQFT-like invariants, however, the above framework seems to be inadequate.

   199 a `fully dualizable object' in some $n+1$-category.

   199 

   200 (The discussion above begins with an object in the $n+1$-category of $n$-categories.

   201 The `sufficiently strong duality' mentioned above corresponds roughly to `fully dualizable'.)

   202 

   203 This formalism successfully captures Turaev-Viro and Reshetikhin-Turaev invariants

   204 (and indeed invariants based on semisimple categories).

   535 \subsubsection{Homotopy colimits}

   532 \subsubsection{Colimits}