215 deduce gluing formulas based on $A_\infty$ tensor products. |
215 deduce gluing formulas based on $A_\infty$ tensor products. |
216 |
216 |
217 \nn{Triangulated categories are important; often calculations are via exact sequences, |
217 \nn{Triangulated categories are important; often calculations are via exact sequences, |
218 and the standard TQFT constructions are quotients, which destroy exactness.} |
218 and the standard TQFT constructions are quotients, which destroy exactness.} |
219 |
219 |
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220 \nn{In many places we omit details; they can be found in MW. |
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221 (Blanket statement in order to avoid too many citations to MW.)} |
220 |
222 |
221 \section{Definitions} |
223 \section{Definitions} |
222 \subsection{$n$-categories} \mbox{} |
224 \subsection{$n$-categories} \mbox{} |
223 |
225 |
224 \nn{rough draft of n-cat stuff...} |
226 \nn{rough draft of n-cat stuff...} |
679 \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]_<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) |
681 \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]_<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) |
680 } |
682 } |
681 \end{equation*} |
683 \end{equation*} |
682 \end{enumerate} |
684 \end{enumerate} |
683 |
685 |
684 Futher, this map is associative, in the sense that the following diagram commutes (up to homotopy). |
686 Further, this map is associative, in the sense that the following diagram commutes (up to homotopy). |
685 \begin{equation*} |
687 \begin{equation*} |
686 \xymatrix{ |
688 \xymatrix{ |
687 \CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor e_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{e_X} \\ |
689 \CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor e_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{e_X} \\ |
688 \CH{X} \tensor \bc_*(X) \ar[r]^{e_X} & \bc_*(X) |
690 \CH{X} \tensor \bc_*(X) \ar[r]^{e_X} & \bc_*(X) |
689 } |
691 } |
690 \end{equation*} |
692 \end{equation*} |
691 \end{thm} |
693 \end{thm} |
692 |
694 |
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695 \nn{if we need to save space, I think this next paragraph could be cut} |
693 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
696 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
694 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
697 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
695 for any homeomorphic pair $X$ and $Y$, |
698 for any homeomorphic pair $X$ and $Y$, |
696 satisfying corresponding conditions. |
699 satisfying corresponding conditions. |
697 |
700 |
698 \nn{Say stuff here!} |
701 \begin{proof}(Sketch.) |
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702 The most convenient way to prove this is to introduce yet another homotopy equivalent version of |
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703 the blob complex, $\cB\cT_*(X)$. |
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704 Blob diagrams have a natural topology, which is ignored by $\bc_*(X)$. |
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705 In $\cB\cT_*(X)$ we take this topology into account, treating the blob diagrams as something |
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706 analogous to a simplicial space (but with cone-product polyhedra replacing simplices). |
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707 More specifically, a generator of $\cB\cT_k(X)$ is an $i$-parameter family of $j$-blob diagrams, with $i+j=k$. |
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708 |
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709 With this alternate version in hand, it is straightforward to prove the theorem. |
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710 The evaluation map $\Homeo(X)\times BD_j(X)\to BD_j(X)$ |
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711 induces a chain map $\CH{X}\ot C_*(BD_j(X))\to C_*(BD_j(X))$ |
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712 and hence a map $e_X: \CH{X} \ot \cB\cT_*(X) \to \cB\cT_*(X)$. |
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713 It is easy to check that $e_X$ thus defined has the desired properties. |
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714 \end{proof} |
699 |
715 |
700 \begin{thm} |
716 \begin{thm} |
701 \label{thm:blobs-ainfty} |
717 \label{thm:blobs-ainfty} |
702 Let $\cC$ be a topological $n$-category. |
718 Let $\cC$ be a topological $n$-category. |
703 Let $Y$ be an $n{-}k$-manifold. |
719 Let $Y$ be an $n{-}k$-manifold. |