175 (i.e.\ TQFTs for which the cylinder 1-category associated to an |
175 (i.e.\ TQFTs for which the cylinder 1-category associated to an |
176 $n{-}1$-manifold $Y$ is semisimple for all $Y$). |
176 $n{-}1$-manifold $Y$ is semisimple for all $Y$). |
177 |
177 |
178 For non-semi-simple TQFTs, this approach is less satisfactory. |
178 For non-semi-simple TQFTs, this approach is less satisfactory. |
179 Our main motivating example (though we will not develop it in this paper) |
179 Our main motivating example (though we will not develop it in this paper) |
180 is the (decapitated) $4{+}1$-dimensional TQFT associated to Khovanov homology. |
180 is the $(4{+}\varepsilon)$-dimensional TQFT associated to Khovanov homology. |
181 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
181 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
182 with a link $L \subset \bd W$. |
182 with a link $L \subset \bd W$. |
183 The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
183 The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
184 %\todo{I'm tempted to replace $A_{Kh}$ with $\cl{Kh}$ throughout this page -S} |
184 %\todo{I'm tempted to replace $A_{Kh}$ with $\cl{Kh}$ throughout this page -S} |
185 |
185 |
207 choices we made along the way. |
207 choices we made along the way. |
208 This is probably not easy to do. |
208 This is probably not easy to do. |
209 |
209 |
210 Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$ |
210 Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$ |
211 which is manifestly invariant. |
211 which is manifestly invariant. |
212 (That is, a definition that does not |
212 In other words, we want a definition that does not |
213 involve choosing a decomposition of $W$. |
213 involve choosing a decomposition of $W$. |
214 After all, one of the virtues of our starting point --- TQFTs via field and local relations --- |
214 After all, one of the virtues of our starting point --- TQFTs via field and local relations --- |
215 is that it has just this sort of manifest invariance.) |
215 is that it has just this sort of manifest invariance. |
216 |
216 |
217 The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient |
217 The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient |
218 \[ |
218 \[ |
219 \text{linear combinations of fields} \;\big/\; \text{local relations} , |
219 \text{linear combinations of fields} \;\big/\; \text{local relations} , |
220 \] |
220 \] |
223 \cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) . |
223 \cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) . |
224 \] |
224 \] |
225 Here $\bc_0$ is linear combinations of fields on $W$, |
225 Here $\bc_0$ is linear combinations of fields on $W$, |
226 $\bc_1$ is linear combinations of local relations on $W$, |
226 $\bc_1$ is linear combinations of local relations on $W$, |
227 $\bc_2$ is linear combinations of relations amongst relations on $W$, |
227 $\bc_2$ is linear combinations of relations amongst relations on $W$, |
228 and so on. We now have a short exact sequence of chain complexes relating resolutions of the link $L$ |
228 and so on. |
|
229 We now have a long exact sequence of chain complexes relating resolutions of the link $L$ |
229 (c.f. Lemma \ref{lem:hochschild-exact} which shows exactness |
230 (c.f. Lemma \ref{lem:hochschild-exact} which shows exactness |
230 with respect to boundary conditions in the context of Hochschild homology). |
231 with respect to boundary conditions in the context of Hochschild homology). |
231 |
232 |
232 |
233 |
233 \subsection{Formal properties} |
234 \subsection{Formal properties} |
423 \end{thm:product} |
424 \end{thm:product} |
424 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
425 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
425 (see \S \ref{ss:product-formula}). |
426 (see \S \ref{ss:product-formula}). |
426 |
427 |
427 Fix a disk-like $n$-category $\cC$, which we'll omit from the notation. |
428 Fix a disk-like $n$-category $\cC$, which we'll omit from the notation. |
428 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ 1-category. |
429 Recall that for any $(n{-}1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ 1-category. |
429 (See Appendix \ref{sec:comparing-A-infty} for the translation between disk-like $A_\infty$ $1$-categories |
430 (See Appendix \ref{sec:comparing-A-infty} for the translation between disk-like $A_\infty$ $1$-categories |
430 and the usual algebraic notion of an $A_\infty$ category.) |
431 and the usual algebraic notion of an $A_\infty$ category.) |
431 |
432 |
432 \newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}} |
433 \newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}} |
433 |
434 |