blob: comparison text/a_inf

equal deleted inserted replaced

-:ba4f86b15ff0
+:c3c8fb292934
 \nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category}
 \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:
+Let $\cE$ be a system of fields (with local relations) and let $\cC_\cE$ be the $A_\infty$
-$$\bc_*(M; \cC) \htpy \cl{\bc_*(pt; \cC)}(M).$$
+$n$-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
 Theorem \ref{thm:product} extends to the case of general fiber bundles
 \[
 	F \to E \to Y .
 \]
-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 $p:D\to Y$
 (c.f. \cite{MR2079378}).
 Call this a $k$-category over $Y$.
 A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$:
 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?}
 \nn{put this later}
 %\nn{need to explain $c$}.
 \item An $A_\infty$ $n{-}k{+}1$-category $\bc(Y)$, defined similarly.
 \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}.
 Let $\cT$ denote the chain complex $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
 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 $J$.
 It follows from Proposition \ref{thm:small-blobs} that $\bc_*(X)$ is homotopic to
 a subcomplex of $G_*$.
 a common refinement.
 The proof that these two maps are inverse to each other is the same as in
 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}
 \label{sec:map-recon}
 \begin{proof}
 The proof is again similar to that of Theorem \ref{thm:product}.
 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
+Recall that
-$j$-fold mapping cylinders, $j \ge 0$.
+the 0-simplices of the homotopy colimit $\cB^\cT(M)$
-So, as an abelian group (but not as a chain complex),
+are a direct sum of chain complexes with the summands indexed by
-\[
-	\cB^\cT(M) = \bigoplus_{j\ge 0} C^j,
-\]
-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 $T$.
+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 $C_*(\Maps(M\to T))$.
 The image of $\psi$ is the subcomplex $G_*\sub C_*(\Maps(M\to T))$ generated by
 families of maps whose support is contained in a disjoint union of balls.
 map $\phi:G_*\to \cB^\cT(M)$ such that $\phi(a)\in D(a)$.
 Furthermore, we may choose $\phi$ such that for all $a$
 \[
 	\phi(a) = (a, K) + r
 \]
-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;}
 \medskip
 \hrule
 \nn{to be continued...}
 \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}

changeset 448	c3c8fb292934
parent 447	ba4f86b15ff0
child 476	4d0ca2fc4f2b