# HG changeset patch # User Kevin Walker # Date 1279834526 21600 # Node ID 07c18e2abd8fafdd393606a08c1135ba0414bb4a # Parent 6a3bc1c10586614b884ed8dc04a950c7a6fdd56a redefine "permissible decomp", and other changes to ntcat.tex; should be read carefully to make sure I didn't introduce inconsistencies diff -r 6a3bc1c10586 -r 07c18e2abd8f text/ncat.tex --- a/text/ncat.tex Thu Jul 22 13:22:34 2010 -0600 +++ b/text/ncat.tex Thu Jul 22 15:35:26 2010 -0600 @@ -251,10 +251,10 @@ \begin{axiom}[Strict associativity] \label{nca-assoc} The composition (gluing) maps above are strictly associative. +Given any splitting of a ball $B$ into smaller balls $B_1,\ldots,B_m$, +any sequence of gluings of the smaller balls yields the same result. \end{axiom} -\nn{should say this means $N$ at a time, not just 3 at a time} - \begin{figure}[!ht] $$\mathfig{.65}{ncat/strict-associativity}$$ \caption{An example of strict associativity.}\label{blah6}\end{figure} @@ -617,11 +617,12 @@ \[ C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . \] -These action maps are required to be associative up to homotopy -\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that +These action maps are required to be associative up to homotopy, +%\nn{iterated homotopy?} +and also compatible with composition (gluing) in the sense that a diagram like the one in Theorem \ref{thm:CH} commutes. -\nn{repeat diagram here?} -\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} +%\nn{repeat diagram here?} +%\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} \end{axiom} We should strengthen the above axiom to apply to families of collar maps. @@ -824,7 +825,7 @@ $n$-category $\cC$ into an $A_\infty$ $n$-category. We think of this as providing a ``free resolution" of the topological $n$-category. -\nn{say something about cofibrant replacements?} +%\nn{say something about cofibrant replacements?} In fact, there is also a trivial, but mostly uninteresting, way to do this: we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, and take $\CD{B}$ to act trivially. @@ -848,7 +849,6 @@ (The topology on this space is induced by ambient isotopy rel boundary. This is homotopy equivalent to a disjoint union of copies $\mathrm{B}\!\Homeo(W')$, where $W'$ runs though representatives of homeomorphism types of such manifolds.) -\nn{check this} \end{example} @@ -859,14 +859,15 @@ boundaries are allowed to meet. Note in particular that the space for $k=1$ contains a copy of $\Diff(B^n)$, namely the embeddings of a ``little" ball with image all of the big ball $B^n$. -\nn{should we warn that the inclusion of this copy of $\Diff(B^n)$ is not a homotopy equivalence?}) +(But note also that this inclusion is not +necessarily a homotopy equivalence.) The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad: by shrinking the little balls (precomposing them with dilations), we see that both operads are homotopic to the space of $k$ framed points in $B^n$. It is easy to see that $n$-fold loop spaces $\Omega^n(T)$ have an action of $\cE\cB_n$. -\nn{add citation for this operad if we can find one} +%\nn{add citation for this operad if we can find one} \begin{example}[$E_n$ algebras] \rm @@ -893,7 +894,7 @@ \nn{should we spell this out?} \nn{Should remark that the associated hocolim for manifolds -is agrees with Lurie's topological chiral homology construction; maybe wait +agrees with Lurie's topological chiral homology construction; maybe wait until next subsection to say that?} Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms @@ -918,11 +919,13 @@ vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. Recall that we can take a plain $n$-category $\cC$ and pass to the ``free resolution", -an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls (recall Example \ref{ex:blob-complexes-of-balls} above). +an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls +(recall Example \ref{ex:blob-complexes-of-balls} above). We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant -for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the same as the original blob complex for $M$ with coefficients in $\cC$. +for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the +same as the original blob complex for $M$ with coefficients in $\cC$. -We will first define the ``cell-decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. +We will first define the ``decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. An $n$-category $\cC$ provides a functor from this poset to the category of sets, and we will define $\cl{\cC}(W)$ as a suitable colimit (or homotopy colimit in the $A_\infty$ case) of this functor. @@ -930,14 +933,22 @@ 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 +Recall (Definition \ref{defn:gluing-decomposition}) that a {\it ball decomposition} of $W$ is a +sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls +$\du_a X_a$. +Abusing notation, we let $X_a$ denote both the ball (component of $M_0$) and +its image in $W$ (which is not necessarily a ball --- parts of $\bd X_a$ may have been glued together). +Define a {\it permissible decomposition} of $W$ to be a map \[ - W = \bigcup_a X_a , + \coprod_a X_a \to W, \] -where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. -\nn{need to define this more carefully} -Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement -of $y$, or write $x \le y$, if each $k$-ball of $y$ is a union of $k$-balls of $x$. +which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. +Roughly, a permissible decomposition is like a ball decomposition where we don't care in which order the balls +are glued up to yield $W$, so long as there is some (non-pathological) way to glue them. + +Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ or $W$, we say that $x$ is a refinement +of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ +with $\du_b Y_b = M_i$ for some $i$. \begin{defn} The category (poset) $\cell(W)$ has objects the permissible decompositions of $W$, @@ -1228,10 +1239,10 @@ \begin{module-axiom}[Strict associativity] The composition and action maps above are strictly associative. +Given any decomposition of a large marked ball into smaller marked and unmarked balls +any sequence of pairwise gluings yields (via composition and action maps) the same result. \end{module-axiom} -\nn{should say that this is multifold, not just 3-fold} - Note that the above associativity axiom applies to mixtures of module composition, action maps and $n$-category composition. See Figure \ref{zzz1b}.