# HG changeset patch # User Kevin Walker # Date 1305345663 25200 # Node ID d8ae97449506d8d2a8307d23e149d19c92e537cc # Parent d75b7bfc44f265262ea6e4ac5e8867f91a7963cf# Parent 0a9adf027f479a52a632d49636e7b2a9f41b8cc2 merging by hand (?) diff -r d75b7bfc44f2 -r d8ae97449506 blob to-do --- a/blob to-do Wed May 11 14:39:21 2011 -0700 +++ b/blob to-do Fri May 13 21:01:03 2011 -0700 @@ -66,3 +66,6 @@ * consider putting conditions for enriched n-cat all in one place +* SCOTT: figure for example 3.1.2 (sin 1/z) + +* SCOTT: add vertical arrow to middle of figure 19 (decomp poset) diff -r d75b7bfc44f2 -r d8ae97449506 blob_changes_v3 --- a/blob_changes_v3 Wed May 11 14:39:21 2011 -0700 +++ b/blob_changes_v3 Fri May 13 21:01:03 2011 -0700 @@ -13,12 +13,16 @@ - changed to pitchfork notation for splittable subsets of fields - added definition of collaring homeomorphism - improved definition of bordism n-category -- fixed definition of a refinement of a ball decomposition (intermediate manifolds should also be disjoiunt unions of balls) +- fixed definition of a refinement of a ball decomposition (intermediate manifolds should also be disjoint unions of balls) - added brief definition of monoidal n-categories - fixed statement of compatibility of product morphisms with gluing - added remark about manifolds which do not admit ball decompositions; restricted product theorem (7.1.1) to apply only to these manifolds - added remarks about categories of defects - clarified that the "cell complexes" in string diagrams are actually a bit more general - added remark to insure that the poset of decompositions is a small category +- rewrote definition of colimit (in "From Balls to Manifolds" subsection) to allow for more general decompositions; also added more details +- + + diff -r d75b7bfc44f2 -r d8ae97449506 text/blobdef.tex --- a/text/blobdef.tex Wed May 11 14:39:21 2011 -0700 +++ b/text/blobdef.tex Fri May 13 21:01:03 2011 -0700 @@ -158,7 +158,7 @@ a manifold. Thus will need to be more careful when speaking of a field $r$ on the complement of the blobs. -\begin{example} +\begin{example} \label{sin1x-example} Consider the four subsets of $\Real^3$, \begin{align*} A & = [0,1] \times [0,1] \times [0,1] \\ diff -r d75b7bfc44f2 -r d8ae97449506 text/ncat.tex --- a/text/ncat.tex Wed May 11 14:39:21 2011 -0700 +++ b/text/ncat.tex Fri May 13 21:01:03 2011 -0700 @@ -936,7 +936,7 @@ 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 $B^n$. (But note also that this inclusion is not -necessarily a homotopy equivalence.) +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 @@ -1001,7 +1001,7 @@ \medskip -We will first define the ``decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. +We will first define the {\it 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. @@ -1037,12 +1037,12 @@ Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, 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 $i$, -and with $M_0,\ldots, M_i$ each being a disjoint union of balls. +and with $M_0, M_1, \ldots, M_i$ each being a disjoint union of balls. \begin{defn} The poset $\cell(W)$ has objects the permissible decompositions of $W$, and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. -See Figure \ref{partofJfig} for an example. +See Figure \ref{partofJfig}. \end{defn} \begin{figure}[t] @@ -1056,25 +1056,86 @@ An $n$-category $\cC$ determines a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets (possibly with additional structure if $k=n$). -Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, +Let $x = \{X_a\}$ be a permissible decomposition of $W$ (i.e.\ object of $\cD(W)$). +We will define $\psi_{\cC;W}(x)$ to be a certain subset of $\prod_a \cC(X_a)$. +Roughly speaking, $\psi_{\cC;W}(x)$ is the subset where the restriction maps from +$\cC(X_a)$ and $\cC(X_b)$ agree whenever some part of $\bd X_a$ is glued to some part of $\bd X_b$. +(Keep in mind that perhaps $a=b$.) +Since we allow decompositions in which the intersection of $X_a$ and $X_b$ might be messy +(see Example \ref{sin1x-example}), we must define $\psi_{\cC;W}(x)$ in a more roundabout way. + +Inductively, we may assume that we have already defined the colimit $\cl\cC(M)$ for $k{-}1$-manifolds $M$. +(To start the induction, we define $\cl\cC(M)$, where $M = \du_a P_a$ is a 0-manifold and each $P_a$ is +a 0-ball, to be $\prod_a \cC(P_a)$.) + +Let $\du_a X_a = M_0\to\cdots\to M_m = W$ be a ball decomposition compatible with $x$. +Let $\bd M_i = N_i \cup Y_i \cup Y'_i$, where $Y_i$ and $Y'_i$ are glued together to produce $M_{i+1}$. +We will define $\psi_{\cC;W}(x)$ be be the subset of $\prod_a \cC(X_a)$ which satisfies a series of conditions +related to the gluings $M_{i-1} \to M_i$, $1\le i \le m$. +By Axiom \ref{nca-boundary}, we have a map +\[ + \prod_a \cC(X_a) \to \cl\cC(\bd M_0) . +\] +The first condition is that the image of $\psi_{\cC;W}(x)$ in $\cl\cC(\bd M_0)$ is splittable +along $\bd Y_0$ and $\bd Y'_0$, and that the restrictions to $\cl\cC(Y_0)$ and $\cl\cC(Y'_0)$ agree +(with respect to the identification of $Y_0$ and $Y'_0$ provided by the gluing map). + +On the subset of $\prod_a \cC(X_a)$ which satisfies the first condition above, we have a restriction +map to $\cl\cC(N_0)$ which we can compose with the gluing map +$\cl\cC(N_0) \to \cl\cC(\bd M_1)$. +The second condition is that the image of $\psi_{\cC;W}(x)$ in $\cl\cC(\bd M_1)$ is splittable +along $\bd Y_1$ and $\bd Y'_1$, and that the restrictions to $\cl\cC(Y_1)$ and $\cl\cC(Y'_1)$ agree +(with respect to the identification of $Y_1$ and $Y'_1$ provided by the gluing map). +The $i$-th condition is defined similarly. + +We define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies the +above conditions for all $i$ and also all +ball decompositions compatible with $x$. +(If $x$ is a nice, non-pathological cell decomposition, then it is easy to see that gluing +compatibility for one ball decomposition implies gluing compatibility for all other ball decompositions. +Rather than try to prove a similar result for arbitrary +permissible decompositions, we instead require compatibility with all ways of gluing up the decomposition.) + +If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. + + +\nn{...} + +\nn{to do: define splittability and restrictions for colimits} + +\noop{ %%%%%%%%%%%%%%%%%%%%%%% +For pedagogical reasons, let us first consider the case of a decomposition $y$ of $W$ +which is a nice, non-pathological cell decomposition. +Then each $k$-ball $X$ of $y$ has its boundary decomposed into $k{-}1$-balls, and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries are splittable along this decomposition. -\begin{defn} -Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. +We can now +define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset \begin{equation} -\label{eq:psi-C} +%\label{eq:psi-C} \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl \end{equation} where the restrictions to the various pieces of shared boundaries amongst the cells $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n{-}1$-cells). If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. + +In general, $y$ might be more general than a cell decomposition +(see Example \ref{sin1x-example}), so we must define $\psi_{\cC;W}$ in a more roundabout way. +\nn{...} + +\begin{defn} +Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. +\nn{...} +If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. \end{defn} +} % end \noop %%%%%%%%%%%%%%%%%%%%%%% + If $k=n$ in the above definition and we are enriching in some auxiliary category, we need to say a bit more. -We can rewrite Equation \ref{eq:psi-C} as +We can rewrite the colimit as \begin{equation} \label{eq:psi-CC} \psi_{\cC;W}(x) \deq \coprod_\beta \prod_a \cC(X_a; \beta) , \end{equation}