# HG changeset patch # User Kevin Walker # Date 1309413978 25200 # Node ID 27e0192b406606a26d5e59830f0057ea0cdcc859 # Parent f38558decd51b471df1d55aceac6852dabd41ef4# Parent b1288cdf16908111c5dc6d2a881d9089ec44fd36 nonsense having to do with resloving conflicts diff -r b1288cdf1690 -r 27e0192b4066 blob to-do --- a/blob to-do Tue Jun 28 18:19:16 2011 -0700 +++ b/blob to-do Wed Jun 29 23:06:18 2011 -0700 @@ -4,13 +4,17 @@ * reconcile splittability with A-inf/families of maps examples * better discussion of systems of fields from disk-like n-cats +** splittability axiom for fields +** topology on fields, topology on morphisms (used in construction of BT) * need to fix fam-o-homeo argument per discussion with Rob * need to change module axioms to follow changes in n-cat axioms; search for and destroy all the "Homeo_\bd"'s, add a v-cone axiom * probably should go through and refer to new splitting axiom when we need to choose refinements etc. - +** in the proof that gluing in dimension < n is injective +** in the proof that D(a) is acyclic +** in the small blobs lemma * framings and duality -- work out what's going on! (alternatively, vague-ify current statement) diff -r b1288cdf1690 -r 27e0192b4066 text/a_inf_blob.tex --- a/text/a_inf_blob.tex Tue Jun 28 18:19:16 2011 -0700 +++ b/text/a_inf_blob.tex Wed Jun 29 23:06:18 2011 -0700 @@ -8,10 +8,10 @@ We will show below in Corollary \ref{cor:new-old} -that when $\cC$ is obtained from a system of fields $\cD$ +that when $\cC$ is obtained from a system of fields $\cE$ as the blob complex of an $n$-ball (see Example \ref{ex:blob-complexes-of-balls}), $\cl{\cC}(M)$ is homotopy equivalent to -our original definition of the blob complex $\bc_*(M;\cD)$. +our original definition of the blob complex $\bc_*(M;\cE)$. %\medskip @@ -51,7 +51,7 @@ First we define a map \[ - \psi: \cl{\cC_F}(Y) \to \bc_*(Y\times F;C) . + \psi: \cl{\cC_F}(Y) \to \bc_*(Y\times F;\cE) . \] On 0-simplices of the hocolimit we just glue together the various blob diagrams on $X_i\times F$ @@ -60,7 +60,7 @@ For simplices of dimension 1 and higher we define the map to be zero. It is easy to check that this is a chain map. -In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;C)$ +In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;\cE)$ and a map \[ \phi: G_* \to \cl{\cC_F}(Y) . @@ -69,9 +69,9 @@ Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding decomposition of $Y\times F$ into the pieces $X_i\times F$. -Let $G_*\sub \bc_*(Y\times F;C)$ be the subcomplex generated by blob diagrams $a$ such that there +Let $G_*\sub \bc_*(Y\times F;\cE)$ be the subcomplex generated by blob diagrams $a$ such that there exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$. -It follows from Lemma \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$ +It follows from Lemma \ref{thm:small-blobs} that $\bc_*(Y\times F; \cE)$ is homotopic to a subcomplex of $G_*$. (If the blobs of $a$ are small with respect to a sufficiently fine cover then their projections to $Y$ are contained in some disjoint union of balls.) @@ -106,7 +106,7 @@ We want to find 1-simplices which connect $K$ and $K'$. We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily the case. -(Consider the $x$-axis and the graph of $y = x^2\sin(1/x)$ in $\r^2$.) +(Consider the $x$-axis and the graph of $y = x^2\sin(1/x)$ in $\r^2$.) \scott{Why the $x^2$ here?} However, we {\it can} find another decomposition $L$ such that $L$ shares common refinements with both $K$ and $K'$. Let $KL$ and $K'L$ denote these two refinements. @@ -412,7 +412,7 @@ \begin{proof} The proof is again similar to that of Theorem \ref{thm:product}. -We begin by constructing chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$. +We begin by constructing a chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$. Recall that the 0-simplices of the homotopy colimit $\cB^\cT(M)$ diff -r b1288cdf1690 -r 27e0192b4066 text/evmap.tex --- a/text/evmap.tex Tue Jun 28 18:19:16 2011 -0700 +++ b/text/evmap.tex Wed Jun 29 23:06:18 2011 -0700 @@ -98,6 +98,7 @@ $\sbc_0(X) = \bc_0(X)$. (This is true for all of the examples presented in this paper.) Accordingly, we define $h_0 = 0$. +\nn{Since we now have an axiom providing this, we should use it. (At present, the axiom is only for morphisms, not fields.)} Next we define $h_1$. Let $b\in C_1$ be a 1-blob diagram. @@ -222,12 +223,12 @@ \item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous. \item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous, where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on -$\bc_0(B)$ comes from the generating set $\BD_0(B)$. +$\bc_0(B)$ comes from the generating set $\BD_0(B)$. \nn{This topology is implicitly part of the data of a system of fields, but never mentioned. It should be!} \end{itemize} We can summarize the above by saying that in the typical continuous family $P\to \BD_k(X)$, $p\mapsto \left(B_i(p), u_i(p), r(p)\right)$, $B_i(p)$ and $r(p)$ are induced by a map -$P\to \Homeo(X)$, with the twig blob labels $u_i(p)$ varying independently. +$P\to \Homeo(X)$, with the twig blob labels $u_i(p)$ varying independently. \nn{``varying independently'' means that \emph{after} you pull back via the family of homeomorphisms to the original twig blob, you see a continuous family of labels, right? We should say this. --- Scott} We note that while we've decided not to allow the blobs $B_i(p)$ to vary independently of the field $r(p)$, if we did allow this it would not affect the truth of the claims we make below. In particular, such a definition of $\btc_*(X)$ would result in a homotopy equivalent complex. @@ -350,7 +351,7 @@ of blob diagrams that are small with respect to $\cU$. (If $f:P \to \BD_k$ is the family then for all $p\in P$ we have that $f(p)$ is a diagram in which the blobs are small.) This is done as in the proof of Lemma \ref{small-blobs-b}; the technique of the proof works in families. -Each such family is homotopic to a sum families which can be a ``lifted" to $\Homeo(X)$. +Each such family is homotopic to a sum of families which can be a ``lifted" to $\Homeo(X)$. That is, $f:P \to \BD_k$ has the form $f(p) = g(p)(b)$ for some $g:P\to \Homeo(X)$ and $b\in \BD_k$. (We are ignoring a complication related to twig blob labels, which might vary independently of $g$, but this complication does not affect the conclusion we draw here.) diff -r b1288cdf1690 -r 27e0192b4066 text/ncat.tex --- a/text/ncat.tex Tue Jun 28 18:19:16 2011 -0700 +++ b/text/ncat.tex Wed Jun 29 23:06:18 2011 -0700 @@ -34,9 +34,8 @@ The axioms for an $n$-category are spread throughout this section. Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, -\ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:vcones} and -\ref{axiom:extended-isotopies}. -For an enriched $n$-category we add \ref{axiom:enriched}. +\ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:extended-isotopies} and \ref{axiom:vcones}. +For an enriched $n$-category we add Axiom \ref{axiom:enriched}. For an $A_\infty$ $n$-category, we replace Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. @@ -207,8 +206,8 @@ We do not insist on surjectivity of the gluing map, since this is not satisfied by all of the examples we are trying to axiomatize. If our $k$-morphisms $\cC(X)$ are labeled cell complexes embedded in $X$ (c.f. Example \ref{ex:traditional-n-categories} below), then a $k$-morphism is -in the image of the gluing map precisely which the cell complex is in general position -with respect to $E$. On the other hand, in categories based on maps to a target space (c.f. Example \ref{ex:maps-to-a-space} below) the gluing map is always surjective +in the image of the gluing map precisely when the cell complex is in general position +with respect to $E$. On the other hand, in categories based on maps to a target space (c.f. Example \ref{ex:maps-to-a-space} below) the gluing map is always surjective. If $S$ is a 0-sphere (the case $k=1$ above), then $S$ can be identified with the {\it disjoint} union of two 0-balls $B_1$ and $B_2$ and the colimit construction $\cl{\cC}(S)$ can be identified @@ -579,7 +578,7 @@ Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. (Keep in mind the important special case where $f$ restricted to $\bd X$ is the identity.) -Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which +Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which act trivially on $\bd b$. Then $f(b) = b$. In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially on @@ -654,7 +653,7 @@ The revised axiom is %\addtocounter{axiom}{-1} -\begin{axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$] +\begin{axiom}[Extended isotopy invariance in dimension $n$] \label{axiom:extended-isotopies} Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. @@ -876,7 +875,7 @@ or more generally an appropriate sort of $\infty$-category, we can modify the extended isotopy axiom \ref{axiom:extended-isotopies} to require that families of homeomorphisms act -and obtain an $A_\infty$ $n$-category. +and obtain what we shall call an $A_\infty$ $n$-category. \noop{ We believe that abstract definitions should be guided by diverse collections @@ -892,7 +891,7 @@ and let $\cJ$ be an $\infty$-functor from topological spaces to $\cS$ (e.g.\ the singular chain functor $C_*$). -\begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] +\begin{axiom}[\textup{\textbf{[$A_\infty$ replacement for Axiom \ref{axiom:extended-isotopies}]}} Families of homeomorphisms act in dimension $n$.] \label{axiom:families} For each pair of $n$-balls $X$ and $X'$ and each pair $c\in \cl{\cC}(\bd X)$ and $c'\in \cl{\cC}(\bd X')$ we have an $\cS$-morphism \[ @@ -913,12 +912,12 @@ We now describe the topology on $\Coll(X; c)$. We retain notation from the above definition of collar map. Each collaring homeomorphism $X \cup (Y\times J) \to X$ determines a map from points $p$ of $\bd X$ to -(possibly zero-width) embedded intervals in $X$ terminating at $p$. +(possibly length zero) embedded intervals in $X$ terminating at $p$. If $p \in Y$ this interval is the image of $\{p\}\times J$. -If $p \notin Y$ then $p$ is assigned the zero-width interval $\{p\}$. +If $p \notin Y$ then $p$ is assigned the length zero interval $\{p\}$. Such collections of intervals have a natural topology, and $\Coll(X; c)$ inherits its topology from this. -Note in particular that parts of the collar are allowed to shrink continuously to zero width. -(This is the real content; if nothing shrinks to zero width then the action of families of collar +Note in particular that parts of the collar are allowed to shrink continuously to zero length. +(This is the real content; if nothing shrinks to zero length then the action of families of collar maps follows from the action of families of homeomorphisms and compatibility with gluing.) The $k=n$ case of Axiom \ref{axiom:morphisms} posits a {\it strictly} associative action of {\it sets} @@ -1000,7 +999,7 @@ \item in the $A_\infty$ case, actions of the topological spaces of homeomorphisms preserving boundary conditions and collar maps (Axiom \ref{axiom:families}). \end{itemize} -The above data must satisfy the following conditions: +The above data must satisfy the following conditions. \begin{itemize} \item The gluing maps are compatible with actions of homeomorphisms and boundary restrictions (Axiom \ref{axiom:composition}). @@ -1118,9 +1117,9 @@ The case $n=d$ captures the $n$-categorical nature of bordisms. The case $n > 2d$ captures the full symmetric monoidal $n$-category structure. \end{example} -\begin{remark} +\begin{rem} Working with the smooth bordism category would require careful attention to either collars, corners or halos. -\end{remark} +\end{rem} %\nn{the next example might be an unnecessary distraction. consider deleting it.} @@ -1344,7 +1343,7 @@ 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 +We will define $\psi_{\cC;W}(x)$ to 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 \[ @@ -1361,7 +1360,7 @@ along $\bd Y_1$ and $\bd Y'_1$, and that the restrictions to $\cl\cC(Y_1)$ and $\cl\cC(Y'_1)$ agree (with respect to the identification of $Y_1$ and $Y'_1$ provided by the gluing map). The $i$-th condition is defined similarly. -Note that these conditions depend on on the boundaries of elements of $\prod_a \cC(X_a)$. +Note that these conditions depend on the boundaries of elements of $\prod_a \cC(X_a)$. We define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies the above conditions for all $i$ and also all @@ -1440,7 +1439,7 @@ of the splitting along $\bd Y$, and this implies that the combined decomposition $\du_{ia} X_{ia}$ is permissible. We can now define the gluing $y_1\bullet y_2$ in the obvious way, and a further application of Axiom \ref{axiom:vcones} -shows that this is independebt of the choices of representatives of $y_i$. +shows that this is independent of the choices of representatives of $y_i$. \medskip @@ -1454,7 +1453,7 @@ \] where $x$ runs through decompositions of $W$, and $\sim$ is the obvious equivalence relation induced by refinement and gluing. -If $\cC$ is enriched over vector spaces and $W$ is an $n$-manifold, +If $\cC$ is enriched over, for example, vector spaces and $W$ is an $n$-manifold, we can take \begin{equation*} \cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K, @@ -2411,7 +2410,7 @@ This will allow us to define $\cS(X; c)$ independently of the choice of $E$. First we must define ``inner product", ``non-degenerate" and ``compatible". -Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ it's mirror image. +Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ its mirror image. (We assume we are working in the unoriented category.) Let $Y\cup\ol{Y}$ denote the decorated $n$-sphere obtained by gluing $Y$ and $\ol{Y}$ along their common boundary. diff -r b1288cdf1690 -r 27e0192b4066 text/tqftreview.tex --- a/text/tqftreview.tex Tue Jun 28 18:19:16 2011 -0700 +++ b/text/tqftreview.tex Wed Jun 29 23:06:18 2011 -0700 @@ -368,7 +368,7 @@ Local relations are subspaces $U(B; c)\sub \cC(B; c)$ of the fields on balls which form an ideal under gluing. Again, we give the examples first. -\addtocounter{prop}{-2} +\addtocounter{subsection}{-2} \begin{example}[contd.] For maps into spaces, $U(B; c)$ is generated by fields of the form $a-b \in \lf(B; c)$, where $a$ and $b$ are maps (fields) which are homotopic rel boundary. @@ -379,6 +379,8 @@ $\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into domain and range. \end{example} +\addtocounter{subsection}{2} +\addtocounter{prop}{-2} These motivate the following definition.