982 The alert reader will have already noticed that our definition of an (ordinary) disk-like $n$-category

   982 The alert reader will have already noticed that our definition of an (ordinary) disk-like $n$-category

   983 is extremely similar to our definition of a system of fields.

   983 is extremely similar to our definition of a system of fields.

   984 There are two differences.

   984 There are two differences.

   985 First, for the $n$-category definition we restrict our attention to balls

   985 First, for the $n$-category definition we restrict our attention to balls

   986 (and their boundaries), while for fields we consider all manifolds.

   986 (and their boundaries), while for fields we consider all manifolds.

   988 invariance in dimension $n$, while in the fields definition we 

   988 invariance in dimension $n$, while in the fields definition we 

   989 instead remember a subspace of local relations which contain differences of isotopic fields. 

   989 instead remember a subspace of local relations which contain differences of isotopic fields. 

   990 (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.)

   990 (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.)

   991 Thus a system of fields and local relations $(\cF,U)$ determines a disk-like $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to

   991 Thus a system of fields and local relations $(\cF,U)$ determines a disk-like $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to

   992 balls and, at level $n$, quotienting out by the local relations:

   992 balls and, at level $n$, quotienting out by the local relations:

  1237 If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point.

  1237 If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point.

  1238 In other words, the $k$-morphisms are trivial for $k<n$.

  1238 In other words, the $k$-morphisms are trivial for $k<n$.

  1239 If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction.

  1239 If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction.

  1240 (Plain colimit, not homotopy colimit.)

  1240 (Plain colimit, not homotopy colimit.)

  1241 Let $J$ be the category whose objects are embeddings of a disjoint union of copies of 

  1241 Let $J$ be the category whose objects are embeddings of a disjoint union of copies of 

  1243 embedded balls into a single larger embedded ball.

  1243 embedded balls into a single larger embedded ball.

  1244 To each object of $J$ we associate $A^{\times m}$ (where $m$ is the number of balls), and

  1244 To each object of $J$ we associate $A^{\times m}$ (where $m$ is the number of balls), and

  1245 to each morphism of $J$ we associate a morphism coming from the $\cE\cB_n$ action on $A$.

  1245 to each morphism of $J$ we associate a morphism coming from the $\cE\cB_n$ action on $A$.

  1246 Alternatively and more simply, we could define $\cC^A(X)$ to be 

  1246 Alternatively and more simply, we could define $\cC^A(X)$ to be 

  1247 $\Diff(B^n\to X)\times A$ modulo the diagonal action of $\Diff(B^n)$.

  1247 $\Diff(B^n\to X)\times A$ modulo the diagonal action of $\Diff(B^n)$.

  1486 \begin{equation*}

  1486 \begin{equation*}

  1487 	\cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K,

  1487 	\cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K,

  1488 \end{equation*}

  1488 \end{equation*}

  1489 where $K$ is the vector space spanned by elements $a - g(a)$, with

  1489 where $K$ is the vector space spanned by elements $a - g(a)$, with

  1490 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x)

  1490 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x)

  1492 

  1492 

  1493 In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit

  1493 In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit

  1494 is more involved.

  1494 is more involved.

  1495 We will describe two different (but homotopy equivalent) versions of the homotopy colimit of $\psi_{\cC;W}$.

  1495 We will describe two different (but homotopy equivalent) versions of the homotopy colimit of $\psi_{\cC;W}$.

  1496 The first is the usual one, which works for any indexing category.

  1496 The first is the usual one, which works for any indexing category.

  1574 The idea of the proof is to produce a similar zig-zag where everything antirefines to the same

  1574 The idea of the proof is to produce a similar zig-zag where everything antirefines to the same

  1575 disjoint union of balls, and then invoke Axiom \ref{nca-assoc} which ensures associativity.

  1575 disjoint union of balls, and then invoke Axiom \ref{nca-assoc} which ensures associativity.

  1576 

  1576 

  1577 Let $z$ be a decomposition of $W$ which is in general position with respect to all of the 

  1577 Let $z$ be a decomposition of $W$ which is in general position with respect to all of the 

  1578 $x_i$'s and $v_i$'s.

  1578 $x_i$'s and $v_i$'s.

  1580 \begin{itemize}

  1580 \begin{itemize}

  1581 \item $x'_i$ antirefines to $x_i$ and $z$;

  1581 \item $x'_i$ antirefines to $x_i$ and $z$;

  1582 \item $v'_i$ antirefines to $x'_i$, $x'_{i-1}$ and $v_i$;

  1582 \item $v'_i$ antirefines to $x'_i$, $x'_{i-1}$ and $v_i$;

  1583 \item $b_i$ is the image of some $b'_i\in \psi(v'_i)$; and

  1583 \item $b_i$ is the image of some $b'_i\in \psi(v'_i)$; and

  1584 \item $a_i$ is the image of some $a'_i\in \psi(x'_i)$, which in turn is the image

  1584 \item $a_i$ is the image of some $a'_i\in \psi(x'_i)$, which in turn is the image