280 \nn{need to expand this; use $\bc_0/\bc_1$ notation (maybe); also introduce

   280 In this subsection we briefly review the construction of a TQFT from a system of fields and local relations.

   281 cylinder categories and gluing formula}

   281 (For more details, see \cite{kw:tqft}.)

   282 

   282 

   283 Given a system of fields and local relations, we define the skein space

   283 Let $W$ be an $n{+}1$-manifold.

   284 $A(Y^n; c)$ to be the space of all finite linear combinations of fields on

   284 We can think of the path integral $Z(W)$ as assigning to each

   285 the $n$-manifold $Y$ modulo local relations.

   285 boundary condition $x\in \cC(\bd W)$ a complex number $Z(W)(x)$.

   286 The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations

   286 In other words, $Z(W)$ lies in $\c^{\lf(\bd W)}$, the vector space of linear

   287 is defined to be the dual of $A(Y; c)$.

   287 maps $\lf(\bd W)\to \c$.

   288 (See \cite{kw:tqft} or xxxx for details.)

   288 

   289 

   289 The locality of the TQFT implies that $Z(W)$ in fact lies in a subspace

   290 \nn{should expand above paragraph}

   290 $Z(\bd W) \sub \c^{\lf(\bd W)}$ defined by local projections.

   291 

   291 The linear dual to this subspace, $A(\bd W) = Z(\bd W)^*$,

   292 The blob complex is in some sense the derived version of $A(Y; c)$.

   292 can be thought of as finite linear combinations of fields modulo local relations.

   293 

   293 (In other words, $A(\bd W)$ is a sort of generalized skein module.)

   294 This is the motivation behind the definition of fields and local relations above.

   295 

   296 In more detail, let $X$ be an $n$-manifold.

   297 %To harmonize notation with the next section, 

   298 %let $\bc_0(X)$ be the vector space of finite linear combinations of fields on $X$, so

   299 %$\bc_0(X) = \lf(X)$.

   300 Define $U(X) \sub \lf(X)$ to be the space of local relations in $\lf(X)$;

   301 $U(X)$ is generated by things of the form $u\bullet r$, where

   302 $u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$.

   303 Define

   304 \[

   305 	A(X) \deq \lf(X) / U(X) .

   306 \]

   307 (The blob complex, defined in the next section, 

   308 is in some sense the derived version of $A(X)$.)

   309 If $X$ has boundary we can similarly define $A(X; c)$ for each 

   310 boundary condition $c\in\cC(\bd X)$.

   311 

   312 The above construction can be extended to higher codimensions, assigning

   313 a $k$-category $A(Y)$ to an $n{-}k$-manifold $Y$, for $0 \le k \le n$.

   314 These invariants fit together via actions and gluing formulas.

   315 We describe only the case $k=1$ below.

   316 (The construction of the $n{+}1$-dimensional part of the theory (the path integral) 

   317 requires that the starting data (fields and local relations) satisfy additional

   318 conditions.

   319 We do not assume these conditions here, so when we say ``TQFT" we mean a decapitated TQFT

   320 that lacks its $n{+}1$-dimensional part.)

   321 

   322 Let $Y$ be an $n{-}1$-manifold.

   323 Define a (linear) 1-category $A(Y)$ as follows.

   324 The objects of $A(Y)$ are $\cC(Y)$.

   325 The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$, where $a$ and $b$ label the two boundary components of the cylinder $Y\times I$.

   326 Composition is given by gluing of cylinders.

   327 

   328 Let $X$ be an $n$-manifold with boundary and consider the collection of vector spaces

   329 $A(X; \cdot) \deq \{A(X; c)\}$ where $c$ ranges through $\cC(\bd X)$.

   330 This collection of vector spaces affords a representation of the category $A(\bd X)$, where

   331 the action is given by gluing a collar $\bd X\times I$ to $X$.

   332 

   333 Given a splitting $X = X_1 \cup_Y X_2$ of a closed $n$-manifold $X$ along an $n{-}1$-manifold $Y$,

   334 we have left and right actions of $A(Y)$ on $A(X_1; \cdot)$ and $A(X_2; \cdot)$.

   335 The gluing theorem for $n$-manifolds states that there is a natural isomorphism

   336 \[

   337 	A(X) \cong A(X_1; \cdot) \otimes_{A(Y)} A(X_2; \cdot) .

   338 \]