10 topological quantum field theory invariant of $M$ associated to $\cC$. |
10 topological quantum field theory invariant of $M$ associated to $\cC$. |
11 (See Proposition \ref{thm:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) |
11 (See Proposition \ref{thm:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) |
12 \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), |
12 \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), |
13 the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. |
13 the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. |
14 (See Theorem \ref{thm:hochschild} and \S \ref{sec:hochschild}.) |
14 (See Theorem \ref{thm:hochschild} and \S \ref{sec:hochschild}.) |
15 \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category, we have |
15 %\item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category, we have |
16 that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
16 %that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
17 on the configuration space of unlabeled points in $M$. (See \S \ref{sec:comm_alg}.) |
17 %on the configuration space of unlabeled points in $M$. (See \S \ref{sec:comm_alg}.) |
18 \item When $\cC$ is $\pi^\infty_{\leq n}(T)$, the $A_\infty$ version of the fundamental $n$-groupoid of |
18 \item When $\cC$ is $\pi^\infty_{\leq n}(T)$, the $A_\infty$ version of the fundamental $n$-groupoid of |
19 the space $T$ (Example \ref{ex:chains-of-maps-to-a-space}), |
19 the space $T$ (Example \ref{ex:chains-of-maps-to-a-space}), |
20 $\bc_*(M; \cC)$ is homotopy equivalent to $C_*(\Maps(M\to T))$, |
20 $\bc_*(M; \cC)$ is homotopy equivalent to $C_*(\Maps(M\to T))$, |
21 the singular chains on the space of maps from $M$ to $T$. |
21 the singular chains on the space of maps from $M$ to $T$. |
22 (See Theorem \ref{thm:map-recon}.) |
22 (See Theorem \ref{thm:map-recon}.) |
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 appendices prove technical results about $\CH{M}$ and |
142 The appendices prove technical results about $\CH{M}$ and |
143 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 |
149 %Throughout the paper we typically prefer concrete categories (vector spaces, chain complexes) |
149 %Throughout the paper we typically prefer concrete categories (vector spaces, chain complexes) |
150 %even when we could work in greater generality (symmetric monoidal categories, model categories, etc.). |
150 %even when we could work in greater generality (symmetric monoidal categories, model categories, etc.). |
151 |
151 |