# HG changeset patch # User Scott Morrison # Date 1323580071 28800 # Node ID ceed680b7f4202a3228e30be8d6520d02e2856c9 # Parent 341c2a09f9a8be937b14ab2665d3341d84769449# Parent b07e2becdd5879fab9e36b84ccfac672c578f4c6 Automated merge with https://tqft.net/hg/blob diff -r b07e2becdd58 -r ceed680b7f42 blob to-do --- a/blob to-do Sat Dec 10 21:07:44 2011 -0800 +++ b/blob to-do Sat Dec 10 21:07:51 2011 -0800 @@ -4,6 +4,9 @@ * add "homeomorphism" spiel befure the first use of "homeomorphism in the intro * maybe also additional homeo warnings in other sections +* Lemma 3.2.3 (\ref{support-shrink}) implicitly assumes embedded (non-self-intersecting) blobs. this can be fixed, of course, but it makes the arument more difficult to understand + + ====== minor/optional ====== @@ -31,8 +34,6 @@ ====== Scott ====== -* SCOTT will go through appendix C.2 and make it better (Schulman's example?) - * SCOTT: review/proof-read recent KW changes, especially colimit section and n-cat axioms diff -r b07e2becdd58 -r ceed680b7f42 blob_changes_v3 --- a/blob_changes_v3 Sat Dec 10 21:07:44 2011 -0800 +++ b/blob_changes_v3 Sat Dec 10 21:07:51 2011 -0800 @@ -37,4 +37,5 @@ - small corrections to proof of product theorem (7.1.1) - added remarks that various homotopy equivalences we construct are well-defined up to a contractible set of choices - clarified that the surgery cylinder operad action is only up to coherent homotopy +- added some details to the construction of a traditional 2-category from a disk-like 2-category diff -r b07e2becdd58 -r ceed680b7f42 text/appendixes/comparing_defs.tex --- a/text/appendixes/comparing_defs.tex Sat Dec 10 21:07:44 2011 -0800 +++ b/text/appendixes/comparing_defs.tex Sat Dec 10 21:07:51 2011 -0800 @@ -530,7 +530,7 @@ Figure \ref{fig:horizontal-compositions-equal}illustrates part of the proof that these four 2-morphisms are equal. Similar arguments show that horizontal composition is associative. \begin{figure}[t] -\begin{equation*} +\begin{align*} \raisebox{-.9cm}{ \begin{tikzpicture} \draw (0,0) .. controls +(1,.8) and +(-1,.8) .. node[above] {$b$} (2.9,0) @@ -544,7 +544,7 @@ .. controls +(-1,-.8) and +(1,-.8) .. node[below] {$c$} (0,0); \draw[->, thick, orange!50!brown] (1.45,-.4)-- node[left, black] {$g$} +(0,.8); \end{tikzpicture}} -\;=\; +\;&=\; \raisebox{-1.9cm}{ \begin{tikzpicture} \draw (0,0) coordinate (p1); @@ -569,11 +569,86 @@ \draw[->, thick, orange!50!brown] (1.45,-1.1)-- node[left, black] {$f$} +(0,.7); \draw[->, thick, orange!50!brown] (4.35,.4)-- node[left, black] {$g$} +(0,.7); \draw[->, thick, blue!75!yellow] (1.5,.78) node[black, above] {$(b\cdot c)\times I$} -- (2.5,0); -\end{tikzpicture}} -\end{equation*} -\begin{equation*} -\mathfig{0.6}{triangle/triangle3b} -\end{equation*} +\end{tikzpicture}} \\ +\;&=\; +\raisebox{-2.1cm}{ +\begin{tikzpicture} + \draw (0,0) coordinate (p1); + \draw (5.8,0) coordinate (p2); + \draw (2.9,0) coordinate (pu); + \draw (2.9,-.9) coordinate (pd); + \begin{scope} + \clip (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu) + .. controls +(.5,-.3) and +(-.6,-.3) .. (p2) + .. controls +(-.6,-.9) and +(.5,0) .. (pd) + .. controls +(-.5,0) and +(.6,-.9) .. (p1); + \foreach \t in {0,.03,...,1} { + \draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); + } + \end{scope} + \draw (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu) + .. controls +(.5,-.3) and +(-.6,-.3) .. (p2) + .. controls +(-.6,-.9) and +(.5,0) .. (pd) + .. controls +(-.5,0) and +(.6,-.9) .. (p1); + \draw (p1) .. controls +(1,1) and +(-1,1) .. (pu); + \draw (p2) .. controls +(-1,1) and +(1,1) .. (pu); + \draw[->, thick, orange!50!brown] (1.45,-0.1)-- node[left, black] {$f$} +(0,.7); + \draw[->, thick, orange!50!brown] (4.35,-0.1)-- node[left, black] {$g$} +(0,.7); + \draw[->, thick, blue!75!yellow] (4.3,-1.5) node[black, below] {$(a\cdot c)\times I$} -- (3.3,-0.5); +\end{tikzpicture}} \\ +\;&=\; +\raisebox{-1.9cm}{ +\begin{tikzpicture}[y=-1cm] + \draw (0,0) coordinate (p1); + \draw (5.8,0) coordinate (p2); + \draw (2.9,.3) coordinate (pu); + \draw (2.9,-.3) coordinate (pd); + \begin{scope} + \clip (p1) .. controls +(.6,.3) and +(-.5,0) .. (pu) + .. controls +(.5,0) and +(-.6,.3) .. (p2) + .. controls +(-.6,-.3) and +(.5,0) .. (pd) + .. controls +(-.5,0) and +(.6,-.3) .. (p1); + \foreach \t in {0,.03,...,1} { + \draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); + } + \end{scope} + \draw (p1) .. controls +(.6,.3) and +(-.5,0) .. (pu) + .. controls +(.5,0) and +(-.6,.3) .. (p2) + .. controls +(-.6,-.3) and +(.5,0) .. (pd) + .. controls +(-.5,0) and +(.6,-.3) .. (p1); + \draw (p1) .. controls +(1,-2) and +(-1,-1) .. (pd); + \draw (p2) .. controls +(-1,2) and +(1,1) .. (pu); + \draw[<-, thick, orange!50!brown] (1.45,-1.1)-- node[left, black] {$f$} +(0,.7); + \draw[<-, thick, orange!50!brown] (4.35,.4)-- node[left, black] {$g$} +(0,.7); + \draw[->, thick, blue!75!yellow] (1.5,.78) node[black, below] {$(a\cdot d)\times I$} -- (2.5,0); +\end{tikzpicture}} \\ +\;&=\; +\raisebox{-1.0cm}{ +\begin{tikzpicture}[y=-1cm] + \draw (0,0) coordinate (p1); + \draw (5.8,0) coordinate (p2); + \draw (2.9,0) coordinate (pu); + \draw (2.9,-.9) coordinate (pd); + \begin{scope} + \clip (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu) + .. controls +(.5,-.3) and +(-.6,-.3) .. (p2) + .. controls +(-.6,-.9) and +(.5,0) .. (pd) + .. controls +(-.5,0) and +(.6,-.9) .. (p1); + \foreach \t in {0,.03,...,1} { + \draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); + } + \end{scope} + \draw (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu) + .. controls +(.5,-.3) and +(-.6,-.3) .. (p2) + .. controls +(-.6,-.9) and +(.5,0) .. (pd) + .. controls +(-.5,0) and +(.6,-.9) .. (p1); + \draw (p1) .. controls +(1,1) and +(-1,1) .. (pu); + \draw (p2) .. controls +(-1,1) and +(1,1) .. (pu); + \draw[<-, thick, orange!50!brown] (1.45,-0.1)-- node[left, black] {$f$} +(0,.7); + \draw[<-, thick, orange!50!brown] (4.35,-0.1)-- node[left, black] {$g$} +(0,.7); + \draw[->, thick, blue!75!yellow] (4.3,-1.5) node[black, above] {$(b\cdot d)\times I$} -- (3.3,-0.5); +\end{tikzpicture}} +\end{align*} \caption{Horizontal composition of 2-morphisms} \label{fzo5} \end{figure} diff -r b07e2becdd58 -r ceed680b7f42 text/basic_properties.tex --- a/text/basic_properties.tex Sat Dec 10 21:07:44 2011 -0800 +++ b/text/basic_properties.tex Sat Dec 10 21:07:51 2011 -0800 @@ -5,7 +5,7 @@ In this section we complete the proofs of Properties \ref{property:disjoint-union}--\ref{property:contractibility}. Throughout the paper, where possible, we prove results using Properties \ref{property:functoriality}--\ref{property:contractibility}, -rather than the actual definition of blob homology. +rather than the actual definition of the blob complex. This allows the possibility of future improvements on or alternatives to our definition. In fact, we hope that there may be a characterization of the blob complex in terms of Properties \ref{property:functoriality}--\ref{property:contractibility}, but at this point we are unaware of one. @@ -112,9 +112,9 @@ } The sum is over all fields $a$ on $Y$ compatible at their ($n{-}2$-dimensional) boundaries with $c$. -``Natural" means natural with respect to the actions of diffeomorphisms. +``Natural" means natural with respect to the actions of homeomorphisms. In degree zero the map agrees with the gluing map coming from the underlying system of fields. \end{prop} This map is very far from being an isomorphism, even on homology. -We fix this deficit in \S\ref{sec:gluing} below. +We eliminate this deficit in \S\ref{sec:gluing} below. diff -r b07e2becdd58 -r ceed680b7f42 text/evmap.tex --- a/text/evmap.tex Sat Dec 10 21:07:44 2011 -0800 +++ b/text/evmap.tex Sat Dec 10 21:07:51 2011 -0800 @@ -50,7 +50,7 @@ \medskip If $b$ is a blob diagram in $\bc_*(X)$, recall from \S \ref{sec:basic-properties} that the {\it support} of $b$, denoted -$\supp(b)$ or $|b|$, to be the union of the blobs of $b$. +$\supp(b)$ or $|b|$, is the union of the blobs of $b$. %For a general $k$-chain $a\in \bc_k(X)$, define the support of $a$ to be the union %of the supports of the blob diagrams which appear in it. More generally, we say that a chain $a\in \bc_k(X)$ is supported on $S$ if @@ -64,14 +64,14 @@ $f$ is supported on $Y$. If $f: M \cup (Y\times I) \to M$ is a collaring homeomorphism -(cf. end of \S \ref{ss:syst-o-fields}), +(cf.\ the end of \S \ref{ss:syst-o-fields}), we say that $f$ is supported on $S\sub M$ if $f(x) = x$ for all $x\in M\setmin S$. \medskip Fix $\cU$, an open cover of $X$. Define the ``small blob complex" $\bc^{\cU}_*(X)$ to be the subcomplex of $\bc_*(X)$ -of all blob diagrams in which every blob is contained in some open set of $\cU$, +generated by blob diagrams such that every blob is contained in some open set of $\cU$, and moreover each field labeling a region cut out by the blobs is splittable into fields on smaller regions, each of which is contained in some open set of $\cU$. @@ -114,7 +114,7 @@ The composition of all the collar maps shrinks $B$ to a ball which is small with respect to $\cU$. Let $\cV_1$ be an auxiliary open cover of $X$, subordinate to $\cU$ and -fine enough that a condition stated later in the proof is satisfied. +fine enough that a condition stated later in this proof is satisfied. Let $b = (B, u, r)$, with $u = \sum a_i$ the label of $B$, and $a_i\in \bc_0(B)$. Choose a sequence of collar maps $\bar{f}_j:B\cup\text{collar}\to B$ satisfying conditions specified at the end of this paragraph. @@ -426,7 +426,7 @@ \eq{ e_{XY} : \CH{X \to Y} \otimes \bc_*(X) \to \bc_*(Y) , } -well-defined up to (coherent) homotopy, +well-defined up to coherent homotopy, such that \begin{enumerate} \item on $C_0(\Homeo(X \to Y)) \otimes \bc_*(X)$ it agrees with the obvious action of diff -r b07e2becdd58 -r ceed680b7f42 text/hochschild.tex --- a/text/hochschild.tex Sat Dec 10 21:07:44 2011 -0800 +++ b/text/hochschild.tex Sat Dec 10 21:07:51 2011 -0800 @@ -212,7 +212,7 @@ (a) the point * is not on the boundary of any blob or (b) there are no labeled points or blob boundaries within distance $\ep$ of *, other than blob boundaries at * itself. -Note that all blob diagrams are in $F_*^\ep$ for $\ep$ sufficiently small. +Note that all blob diagrams are in some $F_*^\ep$ for $\ep$ sufficiently small. Let $b$ be a blob diagram in $F_*^\ep$. Define $f(b)$ to be the result of moving any blob boundary points which lie on * to distance $\ep$ from *. @@ -228,6 +228,7 @@ Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. Extending linearly, we get the desired map $s: J_* \to K_*(C)$. It is easy to check that $s$ is a chain map and $s \circ i = \id$. +What remains is to show that $i \circ s$ is homotopic to the identity. Let $N_\ep$ denote the ball of radius $\ep$ around *. Let $L_*^\ep \sub J_*$ be the subcomplex diff -r b07e2becdd58 -r ceed680b7f42 text/ncat.tex --- a/text/ncat.tex Sat Dec 10 21:07:44 2011 -0800 +++ b/text/ncat.tex Sat Dec 10 21:07:51 2011 -0800 @@ -57,14 +57,16 @@ Still other definitions (see, for example, \cite{MR2094071}) model the $k$-morphisms on more complicated combinatorial polyhedra. -For our definition, we will allow our $k$-morphisms to have any shape, so long as it is +For our definition, we will allow our $k$-morphisms to have {\it any} shape, so long as it is homeomorphic to the standard $k$-ball. Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic to the standard $k$-ball. -By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the + +Below, we will use ``a $k$-ball" to mean any $k$-manifold which is homeomorphic to the standard $k$-ball. -We {\it do not} assume that it is equipped with a -preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. +We {\it do not} assume that such $k$-balls are equipped with a +preferred homeomorphism to the standard $k$-ball. +The same applies to ``a $k$-sphere" below. Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on the boundary), we want a corresponding @@ -240,8 +242,9 @@ .$$ These restriction maps can be thought of as domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. -These restriction maps in fact have their image in the subset $\cC(B_i)\trans E$, -and so to emphasize this we will sometimes write the restriction map as $\cC(X)\trans E \to \cC(B_i)\trans E$. +%%%% the next sentence makes no sense to me, even though I'm probably the one who wrote it -- KW +\noop{These restriction maps in fact have their image in the subset $\cC(B_i)\trans E$, +and so to emphasize this we will sometimes write the restriction map as $\cC(X)\trans E \to \cC(B_i)\trans E$.} Next we consider composition of morphisms. @@ -409,7 +412,7 @@ The need for a strengthened version will become apparent in Appendix \ref{sec:comparing-defs} where we construct a traditional 2-category from a disk-like 2-category. For example, ``half-pinched" products of 1-balls are used to construct weak identities for 1-morphisms -in 2-categories. +in 2-categories (see \S\ref{ssec:2-cats}). We also need fully-pinched products to define collar maps below (see Figure \ref{glue-collar}). Define a {\it pinched product} to be a map