6 \begin{itemize} |
6 \begin{itemize} |
7 \item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. (See Property \ref{property:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) |
7 \item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. (See Property \ref{property:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) |
8 \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. (See Property \ref{property:hochschild} and \S \ref{sec:hochschild}.) |
8 \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. (See Property \ref{property:hochschild} and \S \ref{sec:hochschild}.) |
9 \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have |
9 \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have |
10 that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
10 that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
11 on the configurations space of unlabeled points in $M$. |
11 on the configuration space of unlabeled points in $M$. |
12 %$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ |
12 %$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ |
13 \end{itemize} |
13 \end{itemize} |
14 The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space (replacing quotient of fields by local relations with some sort of resolution), |
14 The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space (replacing quotient of fields by local relations with some sort of resolution), |
15 and for a generalization of Hochschild homology to higher $n$-categories. We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. The blob complex gives us all of these! More detailed motivations are described in \S \ref{sec:motivations}. |
15 and for a generalization of Hochschild homology to higher $n$-categories. We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. The blob complex gives us all of these! More detailed motivations are described in \S \ref{sec:motivations}. |
16 |
16 |