# HG changeset patch # User Scott Morrison # Date 1289519327 28800 # Node ID 20de3d710f77afe1a10c11c9686bfaf96e8b4edd # Parent 26c4d576e155a28077ee5ce228932cb0cfb3d111 writing inconclusively about homotopy colimits, but have to run diff -r 26c4d576e155 -r 20de3d710f77 pnas/pnas.tex --- a/pnas/pnas.tex Tue Nov 09 17:48:16 2010 -0800 +++ b/pnas/pnas.tex Thu Nov 11 15:48:47 2010 -0800 @@ -415,6 +415,7 @@ See Figure \ref{partofJfig} for an example. \end{defn} +This poset in fact has more structure, since we can glue together permissible decompositions of $W_1$ and $W_2$ to obtain a permissible decomposition of $W_1 \sqcup W_2$. An $n$-category $\cC$ determines a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets @@ -443,13 +444,17 @@ \subsubsection{Homotopy colimits} \nn{Motivation: How can we extend an $n$-category from balls to arbitrary manifolds?} -We now define the blob complex $\bc_*(W; \cC)$ of an $n$-manifold $W$ -with coefficients in the $n$-category $\cC$ to be the homotopy colimit -of the functor $\psi_{\cC; W}$ described above. +We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ +with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$ +of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. + +An explicit realization of the homotopy colimit is provided by the simplices of the functor $\psi_{\cC; W}$. That is, $$\clh{\cC}(W) = \DirectSum_{x} \psi_{\cC; W}(x_0)[m],$$ where $x = x_0 \leq \cdots \leq x_m$ is a simplex in $\cell(W)$. The differential acts on \todo{finish this} + +Alternatively, we can take advantage of the product structure on $\cell(W)$ to realize the homotopy colimit as the cone-product polyhedra of the functor $\psi_{\cC;W}$. (A cone-product polyhedra is obtained from a point by successively taking the cone or taking the product with another cone-product polyhedron.) A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. When $\cC$ is a topological $n$-category, the flexibility available in the construction of a homotopy colimit allows -us to give a much more explicit description of the blob complex. +us to give a much more explicit description of the blob complex. We'll write $\bc_*(W; \cC)$ for this more explicit version. We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that @@ -465,9 +470,7 @@ The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with signs given by the ordering. -\todo{Say why this really is the homotopy colimit} - -We now spell this out for some small values of $k$. For $k=0$, the $0$-blob group is simply fields on $W$. For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field that that ball is a null field. The differential simply forgets the ball. Thus we see that $H_0$ of the blob complex is the quotient of fields by null fields. +We now spell this out for some small values of $k$. For $k=0$, the $0$-blob group is simply fields on $W$. For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field to that ball is a null field. The differential simply forgets the ball. Thus we see that $H_0$ of the blob complex is the quotient of fields by fields which are null on some ball. For $k=2$, we have a two types of generators; they each consists of a field $f$ on $W$, and two balls $B_1$ and $B_2$. In the first case, the balls are disjoint, and $f$ restricted to either of the $B_i$ is a null field. In the second case, the balls are properly nested, say $B_1 \subset B_2$, and $f$ restricted to $B_1$ is null. Note that this implies that $f$ restricted to $B_2$ is also null, by the associativity of the gluing operation. This ensures that the differential is well-defined. @@ -630,7 +633,7 @@ Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology. Then \[ - \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). + \bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W). \] \end{thm} The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps diff -r 26c4d576e155 -r 20de3d710f77 pnas/preamble.tex --- a/pnas/preamble.tex Tue Nov 09 17:48:16 2010 -0800 +++ b/pnas/preamble.tex Thu Nov 11 15:48:47 2010 -0800 @@ -12,6 +12,8 @@ \newcommand{\CH}[1]{CH_*(#1)} \newcommand{\cl}[1]{\underrightarrow{#1}} +\newcommand{\clh}[1]{\underrightarrow{#1}_{{}_{{}_{{}_h}}}} + \newcommand{\Set}{\text{\textbf{Set}}} \newcommand{\Vect}{\text{\textbf{Vect}}}