74 The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a |
74 The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a |
75 topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
75 topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
76 |
76 |
77 In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category |
77 In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category |
78 (using a colimit along certain decompositions of a manifold into balls). |
78 (using a colimit along certain decompositions of a manifold into balls). |
79 With this in hand, we freely write $\bc_*(M; \cC)$ to indicate the blob complex of a manifold $M$ |
79 With this in hand, we write $\bc_*(M; \cC)$ to indicate the blob complex of a manifold $M$ |
80 with the system of fields constructed from the $n$-category $\cC$. |
80 with the system of fields constructed from the $n$-category $\cC$. |
81 \nn{KW: I don't think we use this notational convention any more, right?} |
81 %\nn{KW: I don't think we use this notational convention any more, right?} |
82 In \S \ref{sec:ainfblob} we give an alternative definition |
82 In \S \ref{sec:ainfblob} we give an alternative definition |
83 of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). |
83 of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). |
84 Using these definitions, we show how to use the blob complex to ``resolve" any topological $n$-category as an |
84 Using these definitions, we show how to use the blob complex to ``resolve" any topological $n$-category as an |
85 $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. |
85 $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. |
86 We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), |
86 We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), |
125 \draw[->] (C) -- node[above left=3pt] {restrict to \\ standard balls} (tC); |
125 \draw[->] (C) -- node[above left=3pt] {restrict to \\ standard balls} (tC); |
126 \draw[->] (FU.80) -- node[right] {restrict \\ to balls} (C.-80); |
126 \draw[->] (FU.80) -- node[right] {restrict \\ to balls} (C.-80); |
127 \draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Theorem \ref{thm:skein-modules}} (A); |
127 \draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Theorem \ref{thm:skein-modules}} (A); |
128 |
128 |
129 \draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); |
129 \draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); |
130 \draw (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); |
130 \draw[<->] (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); |
131 \end{tikzpicture} |
131 \end{tikzpicture} |
132 |
132 |
133 } |
133 } |
134 \caption{The main gadgets and constructions of the paper.} |
134 \caption{The main gadgets and constructions of the paper.} |
135 \label{fig:outline} |
135 \label{fig:outline} |
137 |
137 |
138 Finally, later sections address other topics. |
138 Finally, later sections address other topics. |
139 Section \S \ref{sec:deligne} gives |
139 Section \S \ref{sec:deligne} gives |
140 a higher dimensional generalization of the Deligne conjecture |
140 a higher dimensional generalization of the Deligne conjecture |
141 (that the little discs operad acts on Hochschild cochains) in terms of the blob complex. |
141 (that the little discs operad acts on Hochschild cochains) in terms of the blob complex. |
142 The appendixes prove technical results about $\CH{M}$ and the ``small blob complex", |
142 The appendices prove technical results about $\CH{M}$ and |
143 and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, |
143 make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, |
144 as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
144 as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
145 Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
145 Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
146 thought of as a topological $n$-category, in terms of the topology of $M$. |
146 thought of as a topological $n$-category, in terms of the topology of $M$. |
147 |
147 |
148 %%%% this is said later in the intro |
148 %%%% this is said later in the intro |
341 \bc_*(X) \ar[d]_{\gl_Y} \\ |
341 \bc_*(X) \ar[d]_{\gl_Y} \\ |
342 \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) |
342 \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) |
343 } |
343 } |
344 \end{equation*} |
344 \end{equation*} |
345 \end{enumerate} |
345 \end{enumerate} |
346 Moreover any such chain map is unique, up to an iterated homotopy. |
346 %Moreover any such chain map is unique, up to an iterated homotopy. |
347 (That is, any pair of homotopies have a homotopy between them, and so on.) |
347 %(That is, any pair of homotopies have a homotopy between them, and so on.) |
348 \nn{revisit this after proof below has stabilized} |
348 %\nn{revisit this after proof below has stabilized} |
349 \end{thm:CH} |
349 \end{thm:CH} |
350 |
350 |
351 \newtheorem*{thm:CH-associativity}{Theorem \ref{thm:CH-associativity}} |
351 \newtheorem*{thm:CH-associativity}{Theorem \ref{thm:CH-associativity}} |
352 |
352 |
353 |
353 |