# HG changeset patch # User Kevin Walker # Date 1279472867 21600 # Node ID c3c8fb29293464b751fb65ede1a6de2beffa630b # Parent ba4f86b15ff03a04eeefbef8980196c51b65c492 done with a-inf section for now diff -r ba4f86b15ff0 -r c3c8fb292934 text/a_inf_blob.tex --- a/text/a_inf_blob.tex Sun Jul 18 08:07:50 2010 -0600 +++ b/text/a_inf_blob.tex Sun Jul 18 11:07:47 2010 -0600 @@ -208,14 +208,18 @@ \medskip +Taking $F$ above to be a point, we obtain the following corollary. + \begin{cor} \label{cor:new-old} -The blob complex of a manifold $M$ with coefficients in a topological $n$-category $\cC$ is homotopic to the homotopy colimit invariant of $M$ defined using the $A_\infty$ $n$-category obtained by applying the blob complex to a point: -$$\bc_*(M; \cC) \htpy \cl{\bc_*(pt; \cC)}(M).$$ +Let $\cE$ be a system of fields (with local relations) and let $\cC_\cE$ be the $A_\infty$ +$n$-category obtained from $\cE$ by taking the blob complex of balls. +Then for all $n$-manifolds $Y$ the old-fashioned and new-fangled blob complexes are +homotopy equivalent: +\[ + \bc^\cE_*(Y) \htpy \cl{\cC_\cE}(Y) . +\] \end{cor} -\begin{proof} -Apply Theorem \ref{thm:product} with the fiber $F$ equal to a point. -\end{proof} \medskip @@ -223,7 +227,7 @@ \[ F \to E \to Y . \] -We outline one approach here and a second in Subsection xxxx. +We outline one approach here and a second in \S \ref{xyxyx}. We can generalize the definition of a $k$-category by replacing the categories of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$ @@ -233,11 +237,11 @@ assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$. Let $\cF_E$ denote this $k$-category over $Y$. We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to -get a chain complex $\cF_E(Y)$. +get a chain complex $\cl{\cF_E}(Y)$. The proof of Theorem \ref{thm:product} goes through essentially unchanged to show that \[ - \bc_*(E) \simeq \cF_E(Y) . + \bc_*(E) \simeq \cl{\cF_E}(Y) . \] \nn{remark further that this still works when the map is not even a fibration?} @@ -276,16 +280,19 @@ \item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked $m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$) or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$). -(See Example \nn{need example for this}.) +(See Example \ref{bc-module-example}.) \end{itemize} +\nn{statement (and proof) is only for case $k=n$; need to revise either above or below; maybe +just say that until we define functors we can't do more} + \begin{thm} \label{thm:gluing} $\bc(X) \simeq \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. \end{thm} \begin{proof} -\nn{for now, just prove $k=0$ case.} +We will assume $k=n$; the other cases are similar. The proof is similar to that of Theorem \ref{thm:product}. We give a short sketch with emphasis on the differences from the proof of Theorem \ref{thm:product}. @@ -294,9 +301,9 @@ Recall that this is a homotopy colimit based on decompositions of the interval $J$. We define a map $\psi:\cT\to \bc_*(X)$. -On filtration degree zero summands it is given +On 0-simplices it is given by gluing the pieces together to get a blob diagram on $X$. -On filtration degree 1 and greater $\psi$ is zero. +On simplices of dimension 1 and greater $\psi$ is zero. The image of $\psi$ is the subcomplex $G_*\sub \bc(X)$ generated by blob diagrams which split over some decomposition of $J$. @@ -313,20 +320,6 @@ Theorem \ref{thm:product}. \end{proof} -\noop{ -Let $\cT$ denote the $n{-}k$-category $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. -Let $D$ be an $n{-}k$-ball. -There is an obvious map from $\cT(D)$ to $\bc_*(D\times X)$. -To get a map in the other direction, we replace $\bc_*(D\times X)$ with a subcomplex -$\cS_*$ which is adapted to a fine open cover of $D\times X$. -For sufficiently small $j$ (depending on the cover), we can find, for each $j$-blob diagram $b$ -on $D\times X$, a decomposition of $J$ such that $b$ splits on the corresponding -decomposition of $D\times X$. -The proof that these two maps are inverse to each other is the same as in -Theorem \ref{thm:product}. -} - - \medskip \subsection{Reconstructing mapping spaces} @@ -361,31 +354,25 @@ We begin by constructing chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$. -Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of -$j$-fold mapping cylinders, $j \ge 0$. -So, as an abelian group (but not as a chain complex), -\[ - \cB^\cT(M) = \bigoplus_{j\ge 0} C^j, -\] -where $C^j$ denotes the new chains introduced by the $j$-fold mapping cylinders. - -Recall that $C^0$ is a direct sum of chain complexes with the summands indexed by +Recall that +the 0-simplices of the homotopy colimit $\cB^\cT(M)$ +are a direct sum of chain complexes with the summands indexed by decompositions of $M$ which have their $n{-}1$-skeletons labeled by $n{-}1$-morphisms of $\cT$. Since $\cT = \pi^\infty_{\leq n}(T)$, this means that the summands are indexed by pairs $(K, \vphi)$, where $K$ is a decomposition of $M$ and $\vphi$ is a continuous -maps from the $n{-}1$-skeleton of $K$ to $T$. +map from the $n{-}1$-skeleton of $K$ to $T$. The summand indexed by $(K, \vphi)$ is \[ \bigotimes_b D_*(b, \vphi), \] where $b$ runs through the $n$-cells of $K$ and $D_*(b, \vphi)$ denotes chains of maps from $b$ to $T$ compatible with $\vphi$. -We can take the product of these chains of maps to get a chains of maps from +We can take the product of these chains of maps to get chains of maps from all of $M$ to $K$. -This defines $\psi$ on $C^0$. +This defines $\psi$ on 0-simplices. -We define $\psi(C^j) = 0$ for $j > 0$. +We define $\psi$ to be zero on $(\ge1)$-simplices. It is not hard to see that this defines a chain map from $\cB^\cT(M)$ to $C_*(\Maps(M\to T))$. @@ -407,49 +394,13 @@ \[ \phi(a) = (a, K) + r \] -where $(a, K) \in C^0$ and $r\in \bigoplus_{j\ge 1} C^j$. +where $(a, K)$ is a 0-simplex and $r$ is a sum of simplices of dimension 1 and greater. It is now easy to see that $\psi\circ\phi$ is the identity on the nose. Another acyclic models argument shows that $\phi\circ\psi$ is homotopic to the identity. (See the proof of Theorem \ref{thm:product} for more details.) \end{proof} -\noop{ -% old proof (just start): -We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$. -We then use Lemma \ref{extension_lemma_c} to show that $g$ induces isomorphisms on homology. - -Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of -$j$-fold mapping cylinders, $j \ge 0$. -So, as an abelian group (but not as a chain complex), -\[ - \cB^\cT(M) = \bigoplus_{j\ge 0} C^j, -\] -where $C^j$ denotes the new chains introduced by the $j$-fold mapping cylinders. - -Recall that $C^0$ is a direct sum of chain complexes with the summands indexed by -decompositions of $M$ which have their $n{-}1$-skeletons labeled by $n{-}1$-morphisms -of $\cT$. -Since $\cT = \pi^\infty_{\leq n}(T)$, this means that the summands are indexed by pairs -$(K, \vphi)$, where $K$ is a decomposition of $M$ and $\vphi$ is a continuous -maps from the $n{-}1$-skeleton of $K$ to $T$. -The summand indexed by $(K, \vphi)$ is -\[ - \bigotimes_b D_*(b, \vphi), -\] -where $b$ runs through the $n$-cells of $K$ and $D_*(b, \vphi)$ denotes -chains of maps from $b$ to $T$ compatible with $\vphi$. -We can take the product of these chains of maps to get a chains of maps from -all of $M$ to $K$. -This defines $g$ on $C^0$. - -We define $g(C^j) = 0$ for $j > 0$. -It is not hard to see that this defines a chain map from -$\cB^\cT(M)$ to $C_*(\Maps(M\to T))$. - -\nn{...} -} - \nn{maybe should also mention version where we enrich over spaces rather than chain complexes;} @@ -461,13 +412,4 @@ \medskip \nn{still to do: general maps} -\todo{} -Various citations we might want to make: -\begin{itemize} -\item \cite{MR2061854} McClure and Smith's review article -\item \cite{MR0420610} May, (inter alia, definition of $E_\infty$ operad) -\item \cite{MR0236922,MR0420609} Boardman and Vogt -\item \cite{MR1256989} definition of framed little-discs operad -\end{itemize} - diff -r ba4f86b15ff0 -r c3c8fb292934 text/ncat.tex --- a/text/ncat.tex Sun Jul 18 08:07:50 2010 -0600 +++ b/text/ncat.tex Sun Jul 18 11:07:47 2010 -0600 @@ -822,6 +822,7 @@ This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. Notice that with $F$ a point, the above example is a construction turning a topological $n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$. +\nn{do we use this notation elsewhere (anymore)?} We think of this as providing a ``free resolution" of the topological $n$-category. \nn{say something about cofibrant replacements?} @@ -1414,6 +1415,15 @@ $\cF(Y)(M)\deq A_\cF((B\times W) \cup (N\times Y); c)$. \end{example} +\begin{example}[Examples from the blob complex] \label{bc-module-example} +\rm +In the previous example, we can instead define +$\cF(Y)(M)\deq \bc_*^\cF((B\times W) \cup (N\times Y); c)$ (when $\dim(M) = n$) +and get a module for the $A_\infty$ $n$-category associated to $\cF$ as in +Example \ref{ex:blob-complexes-of-balls}. +\end{example} + + \begin{example} \rm Suppose $S$ is a topological space, with a subspace $T$.