17 |
17 |
18 It is also worth noting that the original idea for the blob complex came from trying |
18 It is also worth noting that the original idea for the blob complex came from trying |
19 to find a more ``local" description of the Hochschild complex. |
19 to find a more ``local" description of the Hochschild complex. |
20 |
20 |
21 Let $C$ be a *-1-category. |
21 Let $C$ be a *-1-category. |
22 Then specializing the definitions from above to the case $n=1$ we have: |
22 Then specializing the definitions from above to the case $n=1$ we have: \nn{mention that this is dual to the way we think later} \nn{mention that this has the nice side effect of making everything splittable away from the marked points} |
23 \begin{itemize} |
23 \begin{itemize} |
24 \item $\cC(pt) = \ob(C)$ . |
24 \item $\cC(pt) = \ob(C)$ . |
25 \item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$. |
25 \item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$. |
26 Then an element of $\cC(R; c)$ is a collection of (transversely oriented) |
26 Then an element of $\cC(R; c)$ is a collection of (transversely oriented) |
27 points in the interior |
27 points in the interior |
29 The intervals between the points are labeled by objects of $C$, consistent with |
29 The intervals between the points are labeled by objects of $C$, consistent with |
30 the boundary condition $c$ and the domains and ranges of the point labels. |
30 the boundary condition $c$ and the domains and ranges of the point labels. |
31 \item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by |
31 \item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by |
32 composing the morphism labels of the points. |
32 composing the morphism labels of the points. |
33 Note that we also need the * of *-1-category here in order to make all the morphisms point |
33 Note that we also need the * of *-1-category here in order to make all the morphisms point |
34 the same way. |
34 the same way. \nn{Wouldn't it be better to just do the oriented version here? -S} |
35 \item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single |
35 \item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single |
36 point (at some standard location) labeled by $x$. |
36 point (at some standard location) labeled by $x$. |
37 Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the |
37 Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the |
38 form $y - \chi(e(y))$. |
38 form $y - \chi(e(y))$. |
39 Thus we can, if we choose, restrict the blob twig labels to things of this form. |
39 Thus we can, if we choose, restrict the blob twig labels to things of this form. |
128 \cP_i(F_*) & \xrightarrow{\cP_i(\pi)} \cP_i(M) \\ |
128 \cP_i(F_*) & \xrightarrow{\cP_i(\pi)} \cP_i(M) \\ |
129 \intertext{and} |
129 \intertext{and} |
130 \cP_*(F_j) & \xrightarrow{\cP_0(F_j) \onto H_0(\cP_*(F_j))} \coinv(F_j). |
130 \cP_*(F_j) & \xrightarrow{\cP_0(F_j) \onto H_0(\cP_*(F_j))} \coinv(F_j). |
131 \end{align*} |
131 \end{align*} |
132 The cone of each chain map is acyclic. |
132 The cone of each chain map is acyclic. |
133 In the first case, this is because the `rows' indexed by $i$ are acyclic since $\HC_i$ is exact. |
133 In the first case, this is because the `rows' indexed by $i$ are acyclic since $\cP_i$ is exact. |
134 In the second case, this is because the `columns' indexed by $j$ are acyclic, since $F_j$ is free. |
134 In the second case, this is because the `columns' indexed by $j$ are acyclic, since $F_j$ is free. |
135 Because the cones are acyclic, the chain maps are quasi-isomorphisms. |
135 Because the cones are acyclic, the chain maps are quasi-isomorphisms. |
136 Composing one with the inverse of the other, we obtain the desired quasi-isomorphism |
136 Composing one with the inverse of the other, we obtain the desired quasi-isomorphism |
137 $$\cP_*(M) \quismto \coinv(F_*).$$ |
137 $$\cP_*(M) \quismto \coinv(F_*).$$ |
138 |
138 |
234 Let $x \in L_*^\ep$ be a blob diagram. |
234 Let $x \in L_*^\ep$ be a blob diagram. |
235 \nn{maybe add figures illustrating $j_\ep$?} |
235 \nn{maybe add figures illustrating $j_\ep$?} |
236 If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding |
236 If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding |
237 $N_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction |
237 $N_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction |
238 of $x$ to $N_\ep$. |
238 of $x$ to $N_\ep$. |
239 If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, |
239 If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, \nn{I don't think we need to consider sums here} |
240 write $y_i$ for the restriction of $z_i$ to $N_\ep$, and let |
240 write $y_i$ for the restriction of $z_i$ to $N_\ep$, and let |
241 $x_i$ be equal to $x$ on $S^1 \setmin B$, equal to $z_i$ on $B \setmin N_\ep$, |
241 $x_i$ be equal to $x$ on $S^1 \setmin B$, equal to $z_i$ on $B \setmin N_\ep$, |
242 and have an additional blob $N_\ep$ with label $y_i - s(y_i)$. |
242 and have an additional blob $N_\ep$ with label $y_i - s(y_i)$. |
243 Define $j_\ep(x) = \sum x_i$. |
243 Define $j_\ep(x) = \sum x_i$. |
244 |
244 |