31 |
31 |
32 In \S \ref{sec:ainfblob} we explain how to construct a system of fields from a topological $n$-category, and give an alternative definition of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold. Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), in particular the `gluing formula' of Property \ref{property:gluing} below. |
32 In \S \ref{sec:ainfblob} we explain how to construct a system of fields from a topological $n$-category, and give an alternative definition of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold. Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), in particular the `gluing formula' of Property \ref{property:gluing} below. |
33 |
33 |
34 \nn{KW: the previous two paragraphs seem a little awkward to me, but I don't presently have a good idea for fixing them.} |
34 \nn{KW: the previous two paragraphs seem a little awkward to me, but I don't presently have a good idea for fixing them.} |
35 |
35 |
36 Finally, later sections address other topics. Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, thought of as a topological $n$-category, in terms of the topology of $M$. Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves) a higher dimensional generalisation of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. The appendixes prove technical results about $\CH{M}$, and make connections between our definitions of $n$-categories and familar definitions for $n=1$ and $n=2$, as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
36 \tikzstyle{box} = [rectangle, rounded corners, draw,outer sep = 5pt, inner sep = 5pt, line width=0.5pt] |
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37 |
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38 {\center |
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39 |
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40 \begin{tikzpicture}[align=center,line width = 1.5pt] |
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41 \newcommand{\xa}{2} |
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42 \newcommand{\xb}{10} |
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43 \newcommand{\ya}{14} |
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44 \newcommand{\yb}{10} |
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45 \newcommand{\yc}{6} |
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46 |
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47 \node[box] at (\xa,\ya) (C) {$\cC$ \\ a topological \\ $n$-category}; |
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48 \node[box] at (\xb,\ya) (A) {$A(M; \cC)$ \\ the (dual) TQFT \\ Hilbert space}; |
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49 \node[box] at (\xa,\yb) (FU) {$(\cF, \cU)$ \\ fields and\\ local relations}; |
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50 \node[box] at (\xb,\yb) (BC) {$\bc_*(M; \cC)$ \\ the blob complex}; |
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51 \node[box] at (\xa,\yc) (Cs) {$\cC_*$ \\ an $A_\infty$ \\$n$-category}; |
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52 \node[box] at (\xb,\yc) (BCs) {$\bc_*(M; \cC_*)$}; |
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53 |
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54 |
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55 |
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56 \draw[->] (C) -- node[above] {$\displaystyle \colim_{\cell(M)} \cC$} (A); |
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57 \draw[->] (FU) -- node[below] {blob complex \\ for $M$} (BC); |
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58 \draw[->] (Cs) -- node[below] {$\displaystyle \hocolim_{\cell(M)} \cC_*$} (BCs); |
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59 |
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60 \draw[->] (FU) -- node[right=10pt] {$\cF(M)/\cU$} (A); |
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61 |
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62 \draw[->] (C) -- node[left=10pt,align=left] { |
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63 %$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle \cU(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$ |
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64 } (FU); |
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65 \draw[->] (BC) -- node[right] {$H_0$} (A); |
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66 |
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67 \draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); |
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68 \draw (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); |
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69 \end{tikzpicture} |
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70 |
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71 } |
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72 |
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73 Finally, later sections address other topics. Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, thought of as a topological $n$-category, in terms of the topology of $M$. Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves) a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. The appendixes prove technical results about $\CH{M}$, and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
37 |
74 |
38 |
75 |
39 \nn{some more things to cover in the intro} |
76 \nn{some more things to cover in the intro} |
40 \begin{itemize} |
77 \begin{itemize} |
41 \item related: we are being unsophisticated from a homotopy theory point of |
78 \item related: we are being unsophisticated from a homotopy theory point of |