1417 \section{Commutative algebras as $n$-categories}

  1418 

  1419 \nn{this should probably not be a section by itself.  i'm just trying to write down the outline 

  1420 while it's still fresh in my mind.}

  1421 

  1422 If $C$ is a commutative algebra it

  1423 can (and will) also be thought of as an $n$-category with trivial $j$-morphisms for

  1424 $j<n$ and $n$-morphisms are $C$. 

  1425 The goal of this \nn{subsection?} is to compute

  1426 $\bc_*(M^n, C)$ for various commutative algebras $C$.

  1427 

  1428 Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$.

  1429 

  1430 Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$

  1431 unlabeled points in $M$.

  1432 Note that $\Sigma^i(M)$ is a point.

  1433 Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$.

  1434 

  1435 Let $C_*(X)$ denote the singular chain complex of the space $X$.

  1436 

  1437 \begin{prop}

  1438 $\bc_*(M^n, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M))$.

  1439 \end{prop}

  1440 

  1441 \begin{proof}

  1442 To define the chain maps between the two complexes we will use the following lemma:

  1443 

  1444 \begin{lemma}

  1445 Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with

  1446 a basis (e.g.\ blob diagrams or singular simplices).

  1447 For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$

  1448 such that $R(c') \sub R(c)$ whenever $c'$ is a basis element which is part of $\bd c$.

  1449 Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that

  1450 $f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty).

  1451 \end{lemma}

  1452 

  1453 \begin{proof}

  1454 \nn{easy, but should probably write the details eventually}

  1455 \end{proof}

  1456 

  1457 \nn{...}

  1458 

  1459 \end{proof}

  1460 

  1461 

  1462 

  1463 

  1656