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. |
987 Second, in category definition we directly impose isotopy |
987 Second, in the category definition we directly impose isotopy |
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 |
1242 the standard ball $B^n$ into $X$, and who morphisms are given by engulfing some of the |
1242 the standard ball $B^n$ into $X$, and whose morphisms are given by engulfing some of the |
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 $g: \psi_{\cC;W,c}(x) |
1491 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
1491 \to \psi_{\cC;W,c}(y)$ is the value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
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. |
1579 There there decompositions $x'_i$ and $v'_i$ (for all $i$) such that |
1579 There exist decompositions $x'_i$ and $v'_i$ (for all $i$) such that |
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 |