268 other and with $C$ (e.g., producing the functor $M \mapsto \ker(M \tensor C \to M)

   268 other and with $C$ or $\ker(C^{\tensor k} \to C)$ (e.g., producing the functor $M \mapsto \ker(M \tensor C \to M)

   269 \tensor C \tensor \ker(C \tensor M \to M)$) are all still exact.

   269 \tensor C \tensor \ker(C \tensor C \to M)$) are all still exact.

   272 direct sum of this sort of functor. The direct sum is indexed by

   272 direct sum of copies of this sort of functor. The direct sum is indexed by

   273 configurations of nested blobs and positions of labels; for each such configuration, we have one of the above tensor product functors,

   273 configurations of nested blobs and of labels; for each such configuration, we have one of the above tensor product functors,

   274 with the labels of twig blobs corresponding to tensor factors as in \eqref{eq:ker-functor}, and all other labelled points corresponding

   274 with the labels of twig blobs corresponding to tensor factors as in \eqref{eq:ker-functor} or $\ker(C^{\tensor k} \to C)$, and all other labelled points corresponding

   278 \todo{}

   278 We show that $H_0(K_*(M))$ is isomorphic to the coinvariants of $M$.

   279 

   280 We define a map $\ev: K_0(M) \to M$. If $x \in K_0(M)$ has the label $m \in M$ at $*$, and labels $c_i \in C$ at the other labeled points of $S^1$, reading clockwise from $*$,

   281 we set $\ev(x) = m c_1 \cdots c_k$. We can think of this as $\ev : M \tensor C^{\tensor k} \to M$, for each direct summand of $K_0(M)$ indexed by a configuration of labeled points.

   282 There is a quotient map $\pi: M \to \coinv{M}$, and the composition $\pi \compose \ev$ is well-defined on the quotient $H_0(K_*(M))$; if $y \in K_1(M)$, the blob in $y$ either contains $*$ or does not. If it doesn't, then

   283 suppose $y$ has label $m$ at $*$, labels $c_i$ at other labeled points outside the blob, and the field inside the blob is a sum, with the $j$-th term having

   284 labeled points $d_{j,i}$. Then $\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \in \ker(\DirectSum_k C^{\tensor k} \to C)$, and so

   285 $\ev(\bdy y) = 0$, because $$C^{\tensor \ell_1} \tensor \ker(\DirectSum_k C^{\tensor k} \to C) \tensor C^{\tensor \ell_2} \subset \ker(\DirectSum_k C^{\tensor k} \to C).$$

   286 Similarly, if $*$ is contained in the blob, then the blob label is a sum, with the $j$-th term have labelled points $d_{j,i}$ to the left of $*$, $m_j$ at $*$, and $d_{j,i}'$ to the right of $*$,

   287 and there are labels $c_i$ at the labeled points outside the blob. We know that

   288 $$\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \tensor m_j \tensor d_{j,1}' \tensor \cdots \tensor d_{j,k'_j}' \in \ker(\DirectSum_{k,k'} C^{\tensor k} \tensor M \tensor C^{\tensor k'} \tensor \to M),$$

   289 and so

   290 \begin{align*}

   291 \ev(\bdy y) & = \sum_j m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k d_{j,1} \cdots d_{j,k_j} \\

   292             & = \sum_j d_{j,1} \cdots d_{j,k_j} m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k \\

   293             & = 0

   294 \end{align*}

   295 where this time we use the fact that we're mapping to $\coinv{M}$, not just $M$.

   296 

   297 The map $\pi \compose \ev: H_0(K_*(M)) \to \coinv{M}$ is clearly surjective ($\ev$ surjects onto $M$); we now show that it's injective. \todo{}