588 \section{$n=1$ and Hochschild homology}

   588 \section{$n=1$ and Hochschild homology}

   589 

   589 

   590 In this section we analyze the blob complex in dimension $n=1$

   590 In this section we analyze the blob complex in dimension $n=1$

   591 and find that for $S^1$ the homology of the blob complex is the 

   591 and find that for $S^1$ the homology of the blob complex is the 

   592 Hochschild homology of the category (algebroid) that we started with.

   592 Hochschild homology of the category (algebroid) that we started with.

   593 

   595 

   594 Notation: $HB_i(X) = H_i(\bc_*(X))$.

   596 Notation: $HB_i(X) = H_i(\bc_*(X))$.

   595 

   597 

   596 Let us first note that there is no loss of generality in assuming that our system of 

   598 Let us first note that there is no loss of generality in assuming that our system of 

   597 fields comes from a category.

   599 fields comes from a category.

   626 of $R$, each labeled by a morphism of $C$.

   628 of $R$, each labeled by a morphism of $C$.

   627 The intervals between the points are labeled by objects of $C$, consistent with

   629 The intervals between the points are labeled by objects of $C$, consistent with

   628 the boundary condition $c$ and the domains and ranges of the point labels.

   630 the boundary condition $c$ and the domains and ranges of the point labels.

   629 \item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by

   631 \item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by

   630 composing the morphism labels of the points.

   632 composing the morphism labels of the points.

   631 \item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single

   635 \item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single

   632 point (at some standard location) labeled by $x$.

   636 point (at some standard location) labeled by $x$.

   633 Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the 

   637 Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the 

   634 form $y - \chi(e(y))$.

   638 form $y - \chi(e(y))$.

   635 Thus we can, if we choose, restrict the blob twig labels to things of this form.

   639 Thus we can, if we choose, restrict the blob twig labels to things of this form.