# HG changeset patch # User Kevin Walker # Date 1279575515 21600 # Node ID c675b9a331074cdbe65c6da5cbba579a523f8721 # Parent a5d75e0f9229e7377a24bb6c78f18cd3cb5831fa# Parent 54328be726e79435046ec163cd1cc61d53e25f49 Automated merge with https://tqft.net/hg/blob/ diff -r a5d75e0f9229 -r c675b9a33107 diagrams/tempkw/zo2.pdf Binary file diagrams/tempkw/zo2.pdf has changed diff -r a5d75e0f9229 -r c675b9a33107 diagrams/tempkw/zo3.pdf Binary file diagrams/tempkw/zo3.pdf has changed diff -r a5d75e0f9229 -r c675b9a33107 diagrams/tempkw/zo4.pdf Binary file diagrams/tempkw/zo4.pdf has changed diff -r a5d75e0f9229 -r c675b9a33107 text/appendixes/comparing_defs.tex --- a/text/appendixes/comparing_defs.tex Mon Jul 19 12:27:19 2010 -0700 +++ b/text/appendixes/comparing_defs.tex Mon Jul 19 15:38:35 2010 -0600 @@ -108,20 +108,20 @@ \subsection{Plain 2-categories} \label{ssec:2-cats} Let $\cC$ be a topological 2-category. -We will construct a traditional pivotal 2-category. +We will construct from $\cC$ a traditional pivotal 2-category. (The ``pivotal" corresponds to our assumption of strong duality for $\cC$.) We will try to describe the construction in such a way the the generalization to $n>2$ is clear, though this will make the $n=2$ case a little more complicated than necessary. -\nn{Note: We have to decide whether our 2-morphsism are shaped like rectangles or bigons. +Before proceeding, we must decide whether the 2-morphisms of our +pivotal 2-category are shaped like rectangles or bigons. Each approach has advantages and disadvantages. -For better or worse, we choose bigons here.} - -\nn{maybe we should do both rectangles and bigons?} +For better or worse, we choose bigons here. Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k)_E$, where $B^k$ denotes the standard -$k$-ball, which we also think of as the standard bihedron. +$k$-ball, which we also think of as the standard bihedron (a.k.a.\ globe). +(For $k=1$ this is an interval, and for $k=2$ it is a bigon.) Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$ into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$. Recall that the subscript in $\cC(B^k)_E$ means that we consider the subset of $\cC(B^k)$ @@ -136,7 +136,6 @@ on $C^2$ (Figure \ref{fzo1}). Isotopy invariance implies that this is associative. We will define a ``horizontal" composition later. -\nn{maybe no need to postpone?} \begin{figure}[t] \begin{equation*} @@ -146,15 +145,20 @@ \label{fzo1} \end{figure} -Given $a\in C^1$, define $\id_a = a\times I \in C^1$ (pinched boundary). +Given $a\in C^1$, define $\id_a = a\times I \in C^2$ (pinched boundary). Extended isotopy invariance for $\cC$ shows that this morphism is an identity for vertical composition. Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$. We will show that this 1-morphism is a weak identity. This would be easier if our 2-morphisms were shaped like rectangles rather than bigons. + +In showing that identity 1-morphisms have the desired properties, we will +rely heavily on the extended isotopy invariance of 2-morphisms in $\cC$. +This means we are free to add or delete product regions from 2-morphisms. + Let $a: y\to x$ be a 1-morphism. -Define maps $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ +Define 2-morphsims $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ as shown in Figure \ref{fzo2}. \begin{figure}[t] \begin{equation*} @@ -163,10 +167,8 @@ \caption{blah blah} \label{fzo2} \end{figure} -In that figure, the red cross-hatched areas are the product of $x$ and a smaller bigon, -while the remainder is a half-pinched version of $a\times I$. -\nn{the red region is unnecessary; remove it? or does it help? -(because it's what you get if you bigonify the natural rectangular picture)} +As suggested by the figure, these are two different reparameterizations +of a half-pinched version of $a\times I$. We must show that the two compositions of these two maps give the identity 2-morphisms on $a$ and $a\bullet \id_x$, as defined above. Figure \ref{fzo3} shows one case. @@ -177,11 +179,7 @@ \caption{blah blah} \label{fzo3} \end{figure} -In the first step we have inserted a copy of $id(id(x))$ \nn{need better notation for this}. -\nn{also need to talk about (somewhere above) -how this sort of insertion is allowed by extended isotopy invariance and gluing. -Also: maybe half-pinched and unpinched products can be derived from fully pinched -products after all (?)} +In the first step we have inserted a copy of $(x\times I)\times I$. Figure \ref{fzo4} shows the other case. \begin{figure}[t] \begin{equation*} @@ -190,7 +188,7 @@ \caption{blah blah} \label{fzo4} \end{figure} -We first collapse the red region, then remove a product morphism from the boundary, +We identify a product region and remove it. We define horizontal composition of 2-morphisms as shown in Figure \ref{fzo5}. It is not hard to show that this is independent of the arbitrary (left/right) @@ -203,16 +201,8 @@ \label{fzo5} \end{figure} -\nn{need to find a list of axioms for pivotal 2-cats to check} - -\nn{...} +%\nn{need to find a list of axioms for pivotal 2-cats to check} -\medskip -\hrule -\medskip - -\nn{to be continued...} -\medskip \subsection{$A_\infty$ $1$-categories} \label{sec:comparing-A-infty}