# HG changeset patch # User Scott Morrison # Date 1278519450 21600 # Node ID a896ec29425496f783371d8c7b48af9774a0b128 # Parent a571e37cc68d4650435068487eb2391262fb95e5# Parent 257066702f600669f57e678a2fde3784ed599418 Automated merge with https://tqft.net/hg/blob/ diff -r a571e37cc68d -r a896ec294254 text/a_inf_blob.tex --- a/text/a_inf_blob.tex Wed Jul 07 08:47:50 2010 -0600 +++ b/text/a_inf_blob.tex Wed Jul 07 10:17:30 2010 -0600 @@ -41,9 +41,9 @@ 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 $\bc_*(F; C)$ defined by \begin{equation*} -\bc_*(F; C) = \cB_*(B \times F, C). +\bc_*(F; C)(B) = \cB_*(F \times B; C). \end{equation*} -Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the `old-fashioned' +Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the ``old-fashioned'' blob complex for $Y \times F$ with coefficients in $C$ and the ``new-fangled" (i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $\bc_*(F; C)$: \begin{align*} diff -r a571e37cc68d -r a896ec294254 text/ncat.tex --- a/text/ncat.tex Wed Jul 07 08:47:50 2010 -0600 +++ b/text/ncat.tex Wed Jul 07 10:17:30 2010 -0600 @@ -379,7 +379,6 @@ \[ d: \Delta^{k+m}\to\Delta^k . \] -In other words, \nn{each point has a neighborhood blah blah...} (We thank Kevin Costello for suggesting this approach.) Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball, @@ -519,7 +518,7 @@ We start with the plain $n$-category case. -\begin{axiom}[Isotopy invariance in dimension $n$]{\textup{\textbf{[preliminary]}}} +\begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$. @@ -593,7 +592,7 @@ The revised axiom is \addtocounter{axiom}{-1} -\begin{axiom}{\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$.} +\begin{axiom}[\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$.] \label{axiom:extended-isotopies} Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts to the identity on $\bd X$ and isotopic (rel boundary) to the identity. @@ -611,7 +610,7 @@ \addtocounter{axiom}{-1} -\begin{axiom}{\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.} +\begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes \[ C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . @@ -629,7 +628,7 @@ and we can replace the class of all intervals $J$ with intervals contained in $\r$. Having chains on the space of collar maps act gives rise to coherence maps involving weak identities. -We will not pursue this in this draft of the paper. +We will not pursue this in detail here. Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category into a plain $n$-category (enriched over graded groups). @@ -917,7 +916,7 @@ and we will define $\cl{\cC}(W)$ as a suitable colimit (or homotopy colimit in the $A_\infty$ case) of this functor. We'll later give a more explicit description of this colimit. -In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data), +In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-balls with boundary data), then the resulting colimit is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex). Define a {\it permissible decomposition} of $W$ to be a cell decomposition @@ -975,7 +974,7 @@ $\cS$ and the coproduct and product in Equation \ref{eq:psi-CC} should be replaced by the approriate operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect). -Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$. +Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$. \begin{defn}[System of fields functor] \label{def:colim-fields} @@ -1040,7 +1039,7 @@ $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. -\todo{This next sentence is circular: these maps are an axiom, not a consequence of anything. -S} It is easy to see that +It is easy to see that there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps comprise a natural transformation of functors. @@ -1342,10 +1341,10 @@ $(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all such maps modulo homotopies fixed on $\bdy B \setminus N$. This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}. +\end{example} Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and \ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains. -\end{example} \subsection{Modules as boundary labels (colimits for decorated manifolds)} \label{moddecss}