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 \nn{or maybe say here that the complexes are quasi-isomorphic? in general, |
|
594 should perhaps put more emphasis on the complexes and less on the homology.} |
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. |
|
633 Note that we also need the * of *-1-category here in order to make all the morphisms point |
|
634 the same way. |
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. |