185 and to each $(n{+}1)$-manifold $W$ an element of $A(\bd W)^*$. |
185 and to each $(n{+}1)$-manifold $W$ an element of $A(\bd W)^*$. |
186 For the remainder of this paper we will in fact be interested in so-called $(n{+}\epsilon)$-dimensional |
186 For the remainder of this paper we will in fact be interested in so-called $(n{+}\epsilon)$-dimensional |
187 TQFTs, which are slightly weaker structures in that they assign |
187 TQFTs, which are slightly weaker structures in that they assign |
188 invariants to mapping cylinders of homeomorphisms between $n$-manifolds, but not to general $(n{+}1)$-manifolds. |
188 invariants to mapping cylinders of homeomorphisms between $n$-manifolds, but not to general $(n{+}1)$-manifolds. |
189 |
189 |
190 When $k=n-1$ we have a linear 1-category $A(S)$ for each $(n{-}1)$-manifold $S$, |
190 When $k=n{-}1$ we have a linear 1-category $A(S)$ for each $(n{-}1)$-manifold $S$, |
191 and a representation of $A(\bd Y)$ for each $n$-manifold $Y$. |
191 and a representation of $A(\bd Y)$ for each $n$-manifold $Y$. |
192 The TQFT gluing rule in dimension $n$ states that |
192 The TQFT gluing rule in dimension $n$ states that |
193 $A(Y_1\cup_S Y_2) \cong A(Y_1) \ot_{A(S)} A(Y_2)$, |
193 $A(Y_1\cup_S Y_2) \cong A(Y_1) \ot_{A(S)} A(Y_2)$, |
194 where $Y_1$ and $Y_2$ are $n$-manifolds with common boundary $S$. |
194 where $Y_1$ and $Y_2$ are $n$-manifolds with common boundary $S$. |
195 |
195 |
588 Given an isotopy $n$-category $\cC$, |
588 Given an isotopy $n$-category $\cC$, |
589 we will denote its extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$, |
589 we will denote its extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$, |
590 this is defined to be the colimit along $\cell(W)$ of the functor $\psi_{\cC;W}$. |
590 this is defined to be the colimit along $\cell(W)$ of the functor $\psi_{\cC;W}$. |
591 Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity} |
591 Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity} |
592 imply that $\cl{\cC}(X) \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$. |
592 imply that $\cl{\cC}(X) \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$. |
593 Recall that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball, |
593 Suppose that $\cC$ is enriched in vector spaces: this means that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball, |
594 the set $\cC(X;c)$ is a vector space (we assume $\cC$ is enriched in vector spaces). |
594 the set $\cC(X;c)$ is a vector space. |
595 Using this, we note that for $c \in \cl{\cC}(\bdy W)$, |
595 In this case, for $W$ an arbitrary $n$-manifold and $c \in \cl{\cC}(\bdy W)$, |
596 for $W$ an arbitrary $n$-manifold, the set $\cl{\cC}(W;c) = \bdy^{-1} (c)$ inherits the structure of a vector space. |
596 the set $\cl{\cC}(W;c) = \bdy^{-1} (c)$ inherits the structure of a vector space. |
597 These are the usual TQFT skein module invariants on $n$-manifolds. |
597 These are the usual TQFT skein module invariants on $n$-manifolds. |
598 |
598 |
599 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
599 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
600 with coefficients in the $n$-category $\cC$ as the {\it homotopy} colimit along $\cell(W)$ |
600 with coefficients in the $n$-category $\cC$ as the {\it homotopy} colimit along $\cell(W)$ |
601 of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |
601 of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |