blob: comparison pnas/pnas.tex

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 identity morphisms, and special behavior in dimension $n$ (e.g. enrichment
 in some auxiliary category, or strict associativity instead of weak associativity).
 We will treat each of these in turn.
 To motivate our morphism axiom, consider the venerable notion of the Moore loop space
-\nn{need citation -- \S 2.2 of Adams' ``Infinite Loop Spaces''?}.
+\cite[\S 2.2]{MR505692}.
 In the standard definition of a loop space, loops are always parameterized by the unit interval $I = [0,1]$,
 so composition of loops requires a reparameterization $I\cup I \cong I$, and this leads to a proliferation
 of higher associativity relations.
 While this proliferation is manageable for 1-categories (and indeed leads to an elegant theory
 of Stasheff polyhedra and $A_\infty$ categories), it becomes undesirably complex for higher categories.
 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}
+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 $\cell(W)$. The differential acts on $(\bar{x},a)$ (here $a \in \psi_{\cC; W}(x_0)$) as
+$$\bdy (\bar{x},a) = (\bar{x}, \bdy a) + (-1)^{\deg a} \left( (d_0 \bar{x}, g(a)) + \sum_{i=1}^m (-1)^i (d_i \bar{x}, a) \right)$$
-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.
+where $g$ is the gluing map from $x_0$ to $x_1$, and $d_i \bar{x}$ denotes the $i$-th face of the simplex $\bar{x}$.
+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; just as simplices correspond to linear graphs, cone-product polyheda correspond to directed trees.) 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. We'll write $\bc_*(W; \cC)$ for this more explicit version.
+us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$.
 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. 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.
 by $\cC$.
 \begin{equation*}
 H_0(\bc_*(X;\cC)) \iso A_{\cC}(X)
 \end{equation*}
 \end{thm}
-This follows from the fact that the $0$-th homology of a homotopy colimit of \todo{something plain} is the usual colimit, or directly from the explicit description of the blob complex.
+This follows from the fact that the $0$-th homology of a homotopy colimit is the usual colimit, or directly from the explicit description of the blob complex.
 \begin{thm}[Hochschild homology when $X=S^1$]
 \label{thm:hochschild}
 The blob complex for a $1$-category $\cC$ on the circle is
 quasi-isomorphic to the Hochschild complex.
 \begin{thm}[Product formula]
 \label{thm:product}
 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold.
 Let $\cC$ be an $n$-category.
-Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology.
+Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above.
 Then
 \[
 	\bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W).
 \]
 \end{thm}
 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps
 (see \cite[\S7.1]{1009.5025}).
-Fix a topological $n$-category $\cC$, which we'll omit from the notation.
+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.
 \begin{thm}[Gluing formula]
 \label{thm:gluing}
 \mbox{}% <-- gets the indenting right
 An $n$-dimensional surgery cylinder is 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. Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another.
 By the `blob cochains' of a manifold $X$, we mean the $A_\infty$ maps of $\bc_*(X)$ as a $\bc_*(\bdy X)$ $A_\infty$-module.
 \begin{proof}
-We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, and the action of surgeries is just composition of maps of $A_\infty$-modules. We only need to check that the relations of the $n$-SC operad are satisfied. This follows immediately from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and functoriality.
+We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, and the action of surgeries is just composition of maps of $A_\infty$-modules. We only need to check that the relations of the $n$-SC operad are satisfied. This follows immediately from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity.
 \end{proof}
 The little disks operad $LD$ is homotopy equivalent to the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cohains $Hoch^*(C, C)$. The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map
 \[
 	C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}}

changeset 599	ae1ee41f20dd
parent 598	20de3d710f77
child 600	e9032f8dee24