blob: comparison pnas/pnas.tex

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 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$.
 \subsubsection{Colimits}
 Recall that our definition of an $n$-category is essentially a collection of functors
-defined on the categories of homeomorphisms $k$-balls
+defined on the categories of homeomorphisms of $k$-balls
 for $k \leq n$ satisfying certain axioms.
 It is natural to hope to extend such functors to the
 larger categories of all $k$-manifolds (again, with homeomorphisms).
 In fact, the axioms stated above already require such an extension to $k$-spheres for $k<n$.
 The natural construction achieving this is a colimit along the poset of permissible decompositions.
-For an isotopy $n$-category $\cC$,
+Given an isotopy $n$-category $\cC$,
-we will denote the extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$,
+we will denote its extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$,
 this is defined to be the colimit along $\cell(W)$ of the functor $\psi_{\cC;W}$.
 Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity}
 imply that $\cl{\cC}(X)  \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$.
 Recall that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball,
-the set $\cC(X;c)$ is a vector space. Using this, we note that for $c \in \cl{\cC}(\bdy W)$,
+the set $\cC(X;c)$ is a vector space (we assume $\cC$ is enriched in vector spaces).
+Using this, we note that for $c \in \cl{\cC}(\bdy W)$,
 for $W$ an arbitrary $n$-manifold, the set $\cl{\cC}(W;c) = \bdy^{-1} (c)$ inherits the structure of a vector space.
 These are the usual TQFT skein module invariants on $n$-manifolds.
 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 $\cell(W)$
+with coefficients in the $n$-category $\cC$ as the {\it 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, $$\clh{\cC}(W) = \DirectSum_{\bar{x}} \psi_{\cC; W}(x_0)[m],$$
 where $\bar{x} = x_0 \leq \cdots \leq x_m$ is a simplex in $\cell(W)$.
 A cone-product polyhedra is obtained from a point by successively taking the cone or taking the
 product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs,
 cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root,
 and taking product identifies the roots of several trees.
 The ``local homotopy colimit" is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$.
+We further require that any morphism in a directed tree is not expressible as a product.
 The differential acts on $(\bar{x},a)$ both on $a$ and on $\bar{x}$, applying the appropriate gluing map to $a$ when required.
 A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization.
-In the local homotopy colimit we can further require that any composition of morphisms in a directed tree is not expressible as a product.
 %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 which we'll write as $\bc_*(W; \cC)$.
 %\todo{either need to explain why this is the same, or significantly rewrite this section}
 The next few paragraphs describe this in more detail.
 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
 each $B_i$ appears as a connected component of one of the $M_j$.
-Note that this allows the balls to be pairwise either disjoint or nested.
+Note that this forces the balls to be pairwise either disjoint or nested.
 Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$.
 These pieces need not be manifolds, but they do automatically have permissible decompositions.
 The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of
 \begin{itemize}
 The blob complex has several important special cases.
 \begin{thm}[Skein modules]
 \label{thm:skein-modules}
-Suppose $\cC$ is an isotopy $n$-category
+Suppose $\cC$ is an isotopy $n$-category.
 The $0$-th blob homology of $X$ is the usual
 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$
 by $\cC$.
 \begin{equation*}
 H_0(\bc_*(X;\cC)) \iso \cl{\cC}(X)
 \end{thm}
 This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data.
 Note that there is no restriction on the connectivity of $T$ as there is for
 the corresponding result in topological chiral homology \cite[Theorem 3.8.6]{0911.0018}.
-The result was proved in \cite[\S 7.3]{1009.5025}.
+The result is proved in \cite[\S 7.3]{1009.5025}.
 \subsection{Structure of the blob complex}
 \label{sec:structure}
 In the following $\CH{X} = C_*(\Homeo(X))$ is the singular chain complex of the space
 which can be thought of as a free resolution.
 \end{rem}
 This result is described in more detail as Example 6.2.8 of \cite{1009.5025}.
 Fix a topological $n$-category $\cC$, which we'll now omit from notation.
-Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category.
+Recall (Theorem \ref{thm:blobs-ainfty}) that there is associated to
+any $(n{-}1)$-manifold $Y$ an $A_\infty$ category $\bc_*(Y)$.
 \begin{thm}[Gluing formula]
 \label{thm:gluing}
 \mbox{}\vspace{-0.2cm}% <-- gets the indenting right
 \begin{itemize}
 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary,
 the blob complex of $X$ is naturally an
 $A_\infty$ module for $\bc_*(Y)$.
 \item The blob complex of a glued manifold $X\bigcup_Y \selfarrow$ is the $A_\infty$ self-tensor product of
-$\bc_*(X)$ as an $\bc_*(Y)$-bimodule:
+$\bc_*(X)$ as a $\bc_*(Y)$-bimodule:
 \begin{equation*}
 \bc_*(X\bigcup_Y \selfarrow) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow
 \end{equation*}
 \end{itemize}
 \end{thm}
 Let $T_*$ denote the self tensor product of $\bc_*(X)$, which is a homotopy colimit.
 There is a tautological map from the 0-simplices of $T_*$ to $\bc_*(X\bigcup_Y \selfarrow)$,
 and this map can be extended to a chain map on all of $T_*$ by sending the higher simplices to zero.
 Constructing a homotopy inverse to this natural map involves making various choices, but one can show that the
-choices form contractible subcomplexes and apply the theory of acyclic models.
+choices form contractible subcomplexes and apply the acyclic models theorem.
 \end{proof}
 We next describe the blob complex for product manifolds, in terms of the
 blob complexes for the $A_\infty$ $n$-categories constructed as above.
 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above.
 Then
 \[
 	\bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W).
 \]
+That is, the blob complex of $Y\times W$ with coefficients in $\cC$ is homotopy equivalent
+to the blob complex of $W$ with coefficients in $\bc_*(Y;\cC)$.
 \end{thm}
 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps
 (see \cite[\S7.1]{1009.5025}).
 \begin{proof} (Sketch.)
 Recall (Theorem \ref{thm:evaluation}) that chains on the space of mapping cylinders also act on the
 blob complex.
 An $n$-dimensional surgery cylinder is
 defined to be a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}),
 modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism.
-One can associated to this data an $(n{+}1)$-manifold with a foliation by intervals,
+One can associate to this data an $(n{+}1)$-manifold with a foliation by intervals,
 and the relations we impose correspond to homeomorphisms of the $(n{+}1)$-manifolds
 which preserve the foliation.
 Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another.

changeset 659	cc0c2dfe61f3
parent 658	c56a3fe75d1e
child 660	2138fbf11ef8