blob: comparison text/a_inf

equal deleted inserted replaced

-:979fbe9a14e8
+:a02a6158f3bd
 for $X$ an $m$-ball with $m\le k$.
 }
 \nn{need to settle on notation; proof and statement are inconsistent}
-\begin{thm} \label{product_thm}
+\begin{thm} \label{thm:product}
 Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from
 Example \ref{ex:blob-complexes-of-balls} that there is an  $A_\infty$ $k$-category $C^{\times F}$ defined by
 \begin{equation*}
 C^{\times F}(B) = \cB_*(B \times F, C).
 \end{equation*}
 \cB_*(Y \times F, C) & \htpy \cB_*(Y, C^{\times F})
 \end{align*}
 \end{thm}
-\begin{proof}%[Proof of Theorem \ref{product_thm}]
+\begin{proof}
 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}.
 First we define a map
 \[
 	\psi: \bc_*^\cF(Y) \to \bc_*^C(Y\times F) .
 To each generator $(b, \ol{K})$ of $G_*$ we associate the acyclic subcomplex $D(b)$ defined above.
 Both the identity map and $\phi\circ\psi$ are compatible with this
 collection of acyclic subcomplexes, so by the usual MoAM argument these two maps
 are homotopic.
-This concludes the proof of Theorem \ref{product_thm}.
+This concludes the proof of Theorem \ref{thm:product}.
 \end{proof}
 \nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category}
 \medskip
 \begin{cor}
 \label{cor:new-old}
 The new-fangled and old-fashioned blob complexes are homotopic.
 \end{cor}
 \begin{proof}
-Apply Theorem \ref{product_thm} with the fiber $F$ equal to a point.
+Apply Theorem \ref{thm:product} with the fiber $F$ equal to a point.
 \end{proof}
 \medskip
-Theorem \ref{product_thm} extends to the case of general fiber bundles
+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.
 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 $\cF_E(Y)$.
-The proof of Theorem \ref{product_thm} goes through essentially unchanged
+The proof of Theorem \ref{thm:product} goes through essentially unchanged
 to show that
 \[
 	\bc_*(E) \simeq \cF_E(Y) .
 \]
 $\bc(X) \simeq \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
 \end{thm}
 \begin{proof}
 \nn{for now, just prove $k=0$ case.}
-The proof is similar to that of Theorem \ref{product_thm}.
+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{product_thm}.
+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 $\psi:\cT\to \bc_*(X)$.
 over some decomposition of $J$.
 It follows from Proposition \ref{thm:small-blobs} that $\bc_*(X)$ is homotopic to
 a subcomplex of $G_*$.
 Next we define a map $\phi:G_*\to \cT$ using the method of acyclic models.
-As in the proof of Theorem \ref{product_thm}, we assign to a generator $a$ of $G_*$
+As in the proof of Theorem \ref{thm:product}, we assign to a generator $a$ of $G_*$
 an acyclic subcomplex which is (roughly) $\psi\inv(a)$.
 The proof of acyclicity is easier in this case since any pair of decompositions of $J$ have
 a common refinement.
 The proof that these two maps are inverse to each other is the same as in
-Theorem \ref{product_thm}.
+Theorem \ref{thm:product}.
 \end{proof}
-This establishes Property \ref{property:gluing}.
 \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 $\bc_*(D\times X)$.
 $\cS_*$ 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{product_thm}.
+Theorem \ref{thm:product}.
 }
 \medskip
 \subsection{Reconstructing mapping spaces}
+\label{sec:map-recon}
 The next theorem shows how to reconstruct a mapping space from local data.
 Let $T$ be a topological space, let $M$ be an $n$-manifold,
 and recall the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$
 of Example \ref{ex:chains-of-maps-to-a-space}.
 Think of $\pi^\infty_{\leq n}(T)$ as encoding everything you would ever
 want to know about spaces of maps of $k$-balls into $T$ ($k\le n$).
 To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$.
-\begin{thm} \label{thm:map-recon}
+\begin{thm}
+\label{thm:map-recon}
 The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$
 is quasi-isomorphic to singular chains on maps from $M$ to $T$.
 $$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$
 \end{thm}
 \begin{rem}
 is trivial at all but the topmost level.
 Ricardo Andrade also told us about a similar result.
 \end{rem}
 \begin{proof}
-The proof is again similar to that of Theorem \ref{product_thm}.
+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
 $j$-fold mapping cylinders, $j \ge 0$.
 We will define a map $\phi:G_*\to \cB^\cT(M)$ via acyclic models.
 Let $a$ be a generator of $G_*$.
 Define $D(a)$ to be the subcomplex of $\cB^\cT(M)$ generated by all
 pairs $(b, \ol{K})$, where $b$ is a generator appearing in an iterated boundary of $a$
 and $\ol{K}$ is an index of the homotopy colimit $\cB^\cT(M)$.
-(See the proof of Theorem \ref{product_thm} for more details.)
+(See the proof of Theorem \ref{thm:product} for more details.)
 The same proof as of Lemma \ref{lem:d-a-acyclic} shows that $D(a)$ is acyclic.
 By the usual acyclic models nonsense, there is a (unique up to homotopy)
 map $\phi:G_*\to \cB^\cT(M)$ such that $\phi(a)\in D(a)$.
 Furthermore, we may choose $\phi$ such that for all $a$
 \[
 \]
 where $(a, K) \in C^0$ and $r\in \bigoplus_{j\ge 1} C^j$.
 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{product_thm} for more details.)
+(See the proof of Theorem \ref{thm:product} for more details.)
 \end{proof}
 \noop{
 % old proof (just start):
 We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$.

changeset 400	a02a6158f3bd
parent 397	e47dcbf119e7
child 401	a8b8ebcf07ac