author | Kevin Walker <kevin@canyon23.net> |
Thu, 26 Apr 2012 06:57:24 -0600 | |
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parent 972 | 27c075c68816 |
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%!TEX root = ../../blob1.tex |
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\section{Comparing \texorpdfstring{$n$}{n}-category definitions} |
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\label{sec:comparing-defs} |
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In \S\ref{sec:example:traditional-n-categories(fields)} we showed how to construct |
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a disk-like $n$-category from a traditional $n$-category; the morphisms of the |
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disk-like $n$-category are string diagrams labeled by the traditional $n$-category. |
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In this appendix we sketch how to go the other direction, for $n=1$ and 2. |
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The basic recipe, given a disk-like $n$-category $\cC$, is to define the $k$-morphisms |
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of the corresponding traditional $n$-category to be $\cC(B^k)$, where |
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$B^k$ is the {\it standard} $k$-ball. |
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One must then show that the axioms of \S\ref{ss:n-cat-def} imply the traditional $n$-category axioms. |
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One should also show that composing the two arrows (between traditional and disk-like $n$-categories) |
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yields the appropriate sort of equivalence on each side. |
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Since we haven't given a definition for functors between disk-like $n$-categories, we do not pursue this here. |
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We emphasize that we are just sketching some of the main ideas in this appendix --- |
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it falls well short of proving the definitions are equivalent. |
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%\nn{cases to cover: (a) ordinary $n$-cats for $n=1,2$; (b) $n$-cat modules for $n=1$, also 2?; |
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%(c) $A_\infty$ 1-cat; (b) $A_\infty$ 1-cat module?; (e) tensor products?} |
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\subsection{1-categories over \texorpdfstring{$\Set$ or $\Vect$}{Set or Vect}} |
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\label{ssec:1-cats} |
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Given a disk-like $1$-category $\cX$ we construct a $1$-category in the conventional sense, $c(\cX)$. |
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This construction is quite straightforward, but we include the details for the sake of completeness, |
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because it illustrates the role of structures (e.g. orientations, spin structures, etc) |
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on the underlying manifolds, and |
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to shed some light on the $n=2$ case, which we describe in \S \ref{ssec:2-cats}. |
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Let $B^k$ denote the \emph{standard} $k$-ball. |
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Let the objects of $c(\cX)$ be $c(\cX)^0 = \cX(B^0)$ and the morphisms of $c(\cX)$ be $c(\cX)^1 = \cX(B^1)$. |
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The boundary and restriction maps of $\cX$ give domain and range maps from $c(\cX)^1$ to $c(\cX)^0$. |
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Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$. |
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Define composition in $c(\cX)$ to be the induced map $c(\cX)^1\times c(\cX)^1 \to c(\cX)^1$ |
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(defined only when range and domain agree). |
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By isotopy invariance in $\cX$, any other choice of homeomorphism gives the same composition rule. |
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Also by isotopy invariance, composition is strictly associative. |
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Given $a\in c(\cX)^0$, define $\id_a \deq a\times B^1$. |
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By extended isotopy invariance in $\cX$, this has the expected properties of an identity morphism. |
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We have now defined the basic ingredients for the 1-category $c(\cX)$. |
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As we explain below, $c(\cX)$ might have additional structure corresponding to the |
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unoriented, oriented, Spin, $\text{Pin}_+$ or $\text{Pin}_-$ structure on the 1-balls used to define $\cX$. |
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For 1-categories based on unoriented balls, |
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there is a map $\dagger:c(\cX)^1\to c(\cX)^1$ |
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coming from $\cX$ applied to an orientation-reversing homeomorphism (unique up to isotopy) |
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from $B^1$ to itself. |
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(Of course our $B^1$ is unoriented, i.e.\ not equipped with an orientation. |
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We mean the homeomorphism which would reverse the orientation if there were one; |
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$B^1$ is not oriented, but it is orientable.) |
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Topological properties of this homeomorphism imply that |
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$a^{\dagger\dagger} = a$ ($\dagger$ is order 2), $\dagger$ reverses domain and range, and $(ab)^\dagger = b^\dagger a^\dagger$ |
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($\dagger$ is an anti-automorphism). |
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Recall that in this context 0-balls should be thought of as equipped with a germ of a 1-dimensional neighborhood. |
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There is a unique such 0-ball, up to homeomorphism, but it has a non-identity automorphism corresponding to reversing the |
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orientation of the germ. |
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Consequently, the objects of $c(\cX)$ are equipped with an involution, also denoted $\dagger$. |
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If $a:x\to y$ is a morphism of $c(\cX)$ then $a^\dagger: y^\dagger\to x^\dagger$. |
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For 1-categories based on oriented balls, there are no non-trivial homeomorphisms of 0- or 1-balls, and thus no |
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additional structure on $c(\cX)$. |
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For 1-categories based on Spin balls, |
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the nontrivial spin homeomorphism from $B^1$ to itself which covers the identity |
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gives an order 2 automorphism of $c(\cX)^1$. |
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There is a similar involution on the objects $c(\cX)^0$. |
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In the case where there is only one object and we are enriching over complex vector spaces, this |
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is just a super algebra. |
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The even elements are the $+1$ eigenspace of the involution on $c(\cX)^1$, |
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and the odd elements are the $-1$ eigenspace of the involution. |
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For 1-categories based on $\text{Pin}_-$ balls, |
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we have an order 4 antiautomorphism of $c(\cX)^1$. |
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For 1-categories based on $\text{Pin}_+$ balls, |
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we have an order 2 antiautomorphism and also an order 2 automorphism of $c(\cX)^1$, |
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and these two maps commute with each other. |
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In both cases there is a similar map on objects. |
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\noop{ |
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\medskip |
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In the other direction, given a $1$-category $C$ |
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(with objects $C^0$ and morphisms $C^1$) we will construct a disk-like |
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$1$-category $t(C)$. |
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|
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If $X$ is a 0-ball (point), let $t(C)(X) \deq C^0$. |
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If $S$ is a 0-sphere, let $t(C)(S) \deq C^0\times C^0$. |
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If $X$ is a 1-ball, let $t(C)(X) \deq C^1$. |
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Homeomorphisms isotopic to the identity act trivially. |
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If $C$ has extra structure (e.g.\ it's a *-1-category), we use this structure |
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to define the action of homeomorphisms not isotopic to the identity |
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(and get, e.g., an unoriented disk-like 1-category). |
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The domain and range maps of $C$ determine the boundary and restriction maps of $t(C)$. |
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Gluing maps for $t(C)$ are determined by composition of morphisms in $C$. |
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For $X$ a 0-ball, $D$ a 1-ball and $a\in t(C)(X)$, define the product morphism |
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$a\times D \deq \id_a$. |
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It is not hard to verify that this has the desired properties. |
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\medskip |
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The compositions of the constructions above, $$\cX\to c(\cX)\to t(c(\cX))$$ |
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and $$C\to t(C)\to c(t(C)),$$ give back |
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more or less exactly the same thing we started with. |
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As we will see below, for $n>1$ the compositions yield a weaker sort of equivalence. |
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} %end \noop |
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\medskip |
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Similar arguments show that modules for disk-like 1-categories are essentially |
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the same thing as traditional modules for traditional 1-categories. |
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\subsection{Pivotal 2-categories} |
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\label{ssec:2-cats} |
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Let $\cC$ be a disk-like 2-category. |
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We will construct from $\cC$ a traditional pivotal 2-category $D$. |
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(The ``pivotal" corresponds to our assumption of strong duality for $\cC$.) |
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We will try to describe the construction in such a way that the generalization to $n>2$ is clear, |
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though this will make the $n=2$ case a little more complicated than necessary. |
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Before proceeding, we must decide whether the 2-morphisms of our |
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pivotal 2-category are shaped like rectangles or bigons. |
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Each approach has advantages and disadvantages. |
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For better or worse, we choose bigons here. |
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Define the $k$-morphisms $D^k$ of $D$ to be $\cC(B^k) \trans E$, where $B^k$ denotes the standard |
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$k$-ball, which we also think of as the standard bihedron (a.k.a.\ globe). |
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(For $k=1$ this is an interval, and for $k=2$ it is a bigon.) |
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Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$ |
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into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$. |
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Recall that the subscript in $\cC(B^k) \trans E$ means that we consider the subset of $\cC(B^k)$ |
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whose boundary is splittable along $E$. |
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This allows us to define the domain and range of morphisms of $D$ using |
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boundary and restriction maps of $\cC$. |
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Choosing a homeomorphism $B^1\cup B^1 \to B^1$ defines a composition map on $D^1$. |
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This is not associative, but we will see later that it is weakly associative. |
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Choosing a homeomorphism $B^2\cup B^2 \to B^2$ defines a ``vertical" composition map |
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on $D^2$ (Figure \ref{fzo1}). |
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Isotopy invariance implies that this is associative. |
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We will define a ``horizontal" composition later. |
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\begin{figure}[t] |
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\centering |
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\begin{tikzpicture} |
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\newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}} |
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\newcommand{\nsep}{1.8} |
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\node[outer sep=\nsep](A) at (0,0) { |
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\begin{tikzpicture} |
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\draw (0,0) coordinate (p1); |
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\draw (4,0) coordinate (p2); |
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\draw (2,1.2) coordinate (pu); |
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\draw (2,-1.2) coordinate (pd); |
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\draw (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
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\draw (p1)--(p2); |
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\draw (p1) \vertex; |
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\draw (p2) \vertex; |
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\node at (2.1, .44) {$B^2$}; |
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\node at (2.1, -.44) {$B^2$}; |
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\end{tikzpicture} |
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}; |
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\node[outer sep=\nsep](B) at (6,0) { |
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\begin{tikzpicture} |
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\draw (0,0) coordinate (p1); |
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\draw (4,0) coordinate (p2); |
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\draw (2,.6) coordinate (pu); |
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\draw (2,-.6) coordinate (pd); |
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\draw (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
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\draw[help lines, dashed] (p1)--(p2); |
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\draw (p1) \vertex; |
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\draw (p2) \vertex; |
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\node at (2.1,0) {$B^2$}; |
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\end{tikzpicture} |
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}; |
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\draw[->, thick, blue!50!green] (A) -- node[black, above] {$\cong$} (B); |
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\end{tikzpicture} |
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\caption{Vertical composition of 2-morphisms} |
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\label{fzo1} |
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\end{figure} |
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Given $a\in D^1$, define $\id_a = a\times I \in D^2$ (pinched boundary). |
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Extended isotopy invariance for $\cC$ shows that this morphism is an identity for |
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vertical composition. |
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Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$. |
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We will show that this 1-morphism is a weak identity. |
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This would be easier if our 2-morphisms were shaped like rectangles rather than bigons. |
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|
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In showing that identity 1-morphisms have the desired properties, we will |
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rely heavily on the extended isotopy invariance of 2-morphisms in $\cC$. |
934 | 216 |
Extended isotopy invariance implies that adding a product collar to a 2-morphism of $\cC$ has no effect, |
217 |
and by cutting and regluing we can insert (or delete) product regions in the interior of 2-morphisms as well. |
|
955 | 218 |
Figure \ref{fig:product-regions} shows some examples. |
219 |
\begin{figure}[t] |
|
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%$$ |
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%\mathfig{0.5}{triangle/triangle2} |
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%$$ |
958 | 223 |
\begin{align*} |
224 |
\begin{tikzpicture}[baseline] |
|
225 |
\node[draw] (c) at (0,0) [circle through = {(1,0)}] {$f$}; |
|
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\node (d) at (c.east) [circle through = {(0.25,0)}] {}; |
|
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\foreach \n in {1,2} { |
|
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\node (p\n) at (intersection \n of c and d) {}; |
|
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\fill (p\n) circle (2pt); |
|
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} |
|
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\begin{scope}[decoration={brace,amplitude=10,aspect=0.5}] |
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\draw[decorate] (p2.east) -- node[right=2ex] {$a$} (p1.east); |
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\end{scope} |
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\end{tikzpicture} & = |
|
235 |
\begin{tikzpicture}[baseline] |
|
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\node[draw] (c) at (0,0) [circle through = {(1,0)}] {}; |
|
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\begin{scope} |
|
238 |
\path[clip] (c) circle (1); |
|
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\node[draw,dashed] (d) at (c.east) [circle through = {(0.25,0)}] {}; |
|
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\foreach \n in {1,2} { |
|
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\node (p\n) at (intersection \n of c and d) {}; |
|
242 |
} |
|
243 |
\node[left] at (c) {$f$}; |
|
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\path[clip] (d) circle (0.75); |
|
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\foreach \y in {1,0.86,...,-1} { |
|
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\draw[green!50!brown] (0,\y)--(1,\y); |
|
247 |
} |
|
248 |
\end{scope} |
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\draw[->,blue] (1.5,-1) node[below] {$a \times I$} -- (0.75,0); |
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\end{tikzpicture} & |
958 | 251 |
\begin{tikzpicture}[baseline] |
252 |
\node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {}; |
|
253 |
\draw[dashed] (c.north) -- (c.south); |
|
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\node[right=6] at (c) {$g$}; |
|
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\node[left=6] at (c) {$f$}; |
|
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\end{tikzpicture} & = |
|
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\begin{tikzpicture}[baseline] |
|
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\node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {}; |
|
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\node[right=9] at (c) {$g$}; |
|
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\node[left=9] at (c) {$f$}; |
|
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\draw[dashed] (c.north) to[out=-115,in=115] (c.south) to[out=65,in=-65] (c.north); |
|
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\begin{scope} |
|
263 |
\path[clip] (c.north) to[out=-115,in=115] (c.south) to[out=65,in=-65] (c.north); |
|
264 |
\foreach \y in {1,0.86,...,-1} { |
|
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\draw[green!50!brown] (-1,\y)--(1,\y); |
|
266 |
} |
|
267 |
\end{scope} |
|
268 |
\draw[->,blue] (.75,-1.25) node[below] {$a \times I$} -- (0,-0.25); |
|
269 |
\end{tikzpicture} \\ |
|
270 |
\begin{tikzpicture}[baseline] |
|
271 |
\node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {}; |
|
272 |
\draw[dashed] (c.north) -- (c.south); |
|
273 |
\node[right=18] at (c) {$g$}; |
|
274 |
\node[left=10] at (c) {$f$}; |
|
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\fill (0,0.4) node (p1) {} circle (2pt); |
|
276 |
\fill (0,-0.4) node (p2) {} circle (2pt); |
|
277 |
\begin{scope}[decoration={brace,amplitude=5,aspect=0.5}] |
|
278 |
\draw[decorate] (p1.east) -- node[right=0.5ex] {\scriptsize $a$} (p2.east); |
|
279 |
\end{scope} |
|
280 |
\end{tikzpicture} & = |
|
281 |
\begin{tikzpicture}[baseline] |
|
282 |
\node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {}; |
|
283 |
\node[draw,dashed] (d) at (0,0) [circle, minimum height=1cm,minimum width=1cm] {}; |
|
284 |
\draw[dashed] (c.north) -- (d.north) (d.south) -- (c.south); |
|
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\node[right=18] at (c) {$g$}; |
|
286 |
\node[left=18] at (c) {$f$}; |
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\draw[->,blue] (.75,-1.25) node[below] {$a \times I$} -- (0,-0.25); |
958 | 288 |
\clip (0,0) circle (0.5cm); |
289 |
\foreach \y in {1,0.86,...,-1} { |
|
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\draw[green!50!brown] (-1,\y)--(1,\y); |
|
291 |
} |
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\end{tikzpicture} & |
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\begin{tikzpicture}[baseline] |
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\begin{scope} |
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\clip (-1.3,-2) rectangle (0,2); |
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\node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {}; |
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\end{scope} |
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\begin{scope} |
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\clip (1,-2) rectangle (0,2); |
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300 |
\node[draw] (d) at (0,-0.4) [ellipse, minimum height=1.2cm, minimum width=1.5cm] {}; |
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\end{scope} |
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\draw (c.north) -- (d.north); |
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\draw[dashed] (d.north) -- (d.south); |
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\node[right=8,below=4] at (c) {$g$}; |
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\node[left=10] at (c) {$f$}; |
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\end{tikzpicture} & = |
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307 |
\begin{tikzpicture}[baseline] |
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308 |
\begin{scope} |
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309 |
\clip (-1.3,-2) rectangle (0,2); |
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310 |
\node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {}; |
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\end{scope} |
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312 |
\draw[dashed] (c.north) to[out=-120,in=120] (c.south); |
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313 |
\draw[dashed] (c.south) to[out=60,in=-90] (0.35,0) node (m) {}; |
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314 |
\draw (m.center) to[out=90,in=-60] (c.north); |
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315 |
\begin{scope} |
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|
316 |
\clip (c.north) to[out=-120,in=120] (c.south) to[out=60,in=-90] (m) to[out=90,in=-60] (c.north); |
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317 |
\foreach \y in {1,0.86,...,-1} { |
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318 |
\draw[green!50!brown] (-1,\y)--(1,\y); |
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|
319 |
} |
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320 |
\end{scope} |
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|
321 |
\draw (m.center) .. controls +(1,0) and +(1,0) .. (c.south); |
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322 |
\node[right=15,below=8] at (c) {$g$}; |
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|
323 |
\node[left=12] at (c) {$f$}; |
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324 |
\draw[->,blue] (.65,-1.25) node[below] {$a \times I$} -- (0,-0.25); |
960
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|
325 |
\end{tikzpicture} |
958 | 326 |
\end{align*} |
955 | 327 |
\caption{Examples of inserting or deleting product regions.} |
328 |
\label{fig:product-regions} |
|
329 |
\end{figure} |
|
934 | 330 |
|
457
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|
331 |
|
201 | 332 |
Let $a: y\to x$ be a 1-morphism. |
457
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|
333 |
Define 2-morphsims $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ |
125 | 334 |
as shown in Figure \ref{fzo2}. |
126 | 335 |
\begin{figure}[t] |
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|
336 |
\centering |
498
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|
337 |
\begin{tikzpicture} |
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|
338 |
\newcommand{\rr}{6} |
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339 |
\newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}} |
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|
340 |
\newcommand{\namedvertex}[1]{node[circle,fill=black,inner sep=1pt] (#1) {}} |
498
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341 |
|
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|
342 |
\node(A) at (0,0) { |
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|
343 |
\begin{tikzpicture} |
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|
344 |
\node[red,left] at (0,0) {$y$}; |
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|
345 |
\draw (0,0) \vertex arc (-120:-105:\rr) node[red,below] {$a$} arc(-105:-90:\rr) \vertex node[red,below](x2) {$x$}; |
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|
346 |
\draw (0,0) \vertex arc (120:105:\rr) node[red,above] {$a$} arc (105:90:\rr) \vertex node[red,above](x1) {$x$} -- (x2); |
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|
347 |
\begin{scope} |
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348 |
\path[clip] (0,0) arc (-120:-60:\rr) arc (60:120:\rr); |
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349 |
\foreach \x in {0,0.24,...,3} { |
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|
350 |
\draw[green!50!brown] (\x,1) -- (\x,-1); |
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|
351 |
} |
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352 |
\end{scope} |
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353 |
\draw[red, decorate,decoration={brace,amplitude=5pt}] ($(x1)+(0.2,-0.2)$) -- ($(x2)+(0.2,0.2)$) node[midway, xshift=0.7cm] {$x \times I$}; |
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354 |
\end{tikzpicture} |
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|
355 |
}; |
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|
356 |
|
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|
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\node(B) at (-4,-4) { |
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|
358 |
\begin{tikzpicture} |
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|
359 |
\node[red,left] at (0,0) {$y$}; |
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|
360 |
\draw (0,0) \vertex |
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|
361 |
arc (120:105:\rr) node[red,above] {$a$} |
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|
362 |
arc (105:90:\rr) node[red,above] {$x$} \namedvertex{x1}; |
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% arc (90:75:\rr) node[red,above] {$x \times I$}; |
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|
364 |
% arc (75:60:\rr) \vertex node[red,right] {$x$} |
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% arc (-60:-90:\rr) node[red,below] {$a$} |
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366 |
% arc (-90:-120:\rr); |
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367 |
\draw (0,0) |
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|
368 |
arc (-120:-90:\rr) node[red,below] {$a$} |
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arc (-90:-61:\rr) \namedvertex{x2} node[red,right] {$x$}; |
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\draw (x1) -- node[red, above=3pt] {$x \times I$} (x2); |
498
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\begin{scope} |
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372 |
\path[clip] (0,0) arc (-120:-60:\rr) arc (60:120:\rr); |
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\foreach \x in {0,0.48,...,9} { |
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\draw[green!50!brown] (\x/4,1) -- (\x,-1); |
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375 |
} |
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376 |
\end{scope} |
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377 |
\end{tikzpicture} |
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378 |
}; |
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|
379 |
|
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
380 |
\node(C) at (4,-4) { |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
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diff
changeset
|
381 |
\begin{tikzpicture}[y=-1cm] |
b98790f0282e
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Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
382 |
\node[red,left] at (0,0) {$y$}; |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
383 |
\draw (0,0) \vertex |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
384 |
arc (120:105:\rr) node[red,below] {$a$} |
503
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
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498
diff
changeset
|
385 |
arc (105:90:\rr) node[red,below] {$x$} \namedvertex{x1}; |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
386 |
% arc (90:75:\rr) node[red,below] {$x \times I$} |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
387 |
% arc (75:60:\rr) \vertex node[red,right] {$x$} |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
388 |
% arc (-60:-90:\rr) node[red,above] {$a$} |
c221d8331f30
straight lines in figure 42
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parents:
498
diff
changeset
|
389 |
% arc (-90:-120:\rr); |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
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diff
changeset
|
390 |
\draw (0,0) |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
391 |
arc (-120:-90:\rr) node[red,above] {$a$} |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
392 |
arc (-90:-61:\rr) \namedvertex{x2} node[red,right] {$x$}; |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
393 |
\draw (x1) -- node[red, below=3pt] {$x \times I$} (x2); |
498
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
394 |
\begin{scope} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
395 |
\path[clip] (0,0) arc (-120:-60:\rr) arc (60:120:\rr); |
b98790f0282e
diagram for producing weak identities
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parents:
481
diff
changeset
|
396 |
\foreach \x in {0,0.48,...,9} { |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
397 |
\draw[green!50!brown] (\x/4,1) -- (\x,-1); |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
398 |
} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
399 |
\end{scope} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
400 |
\end{tikzpicture} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
401 |
}; |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
402 |
|
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
403 |
\draw[->] (A) -- (B); |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
404 |
\draw[->] (A) -- (C); |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
405 |
\end{tikzpicture} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
406 |
\caption{Producing weak identities from half pinched products} |
126 | 407 |
\label{fzo2} |
408 |
\end{figure} |
|
457
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
409 |
As suggested by the figure, these are two different reparameterizations |
934 | 410 |
of a half-pinched version of $a\times I$ |
411 |
(i.e.\ two different homeomorphisms from the half-pinched $I\times I$ to the standard bigon). |
|
125 | 412 |
We must show that the two compositions of these two maps give the identity 2-morphisms |
413 |
on $a$ and $a\bullet \id_x$, as defined above. |
|
414 |
Figure \ref{fzo3} shows one case. |
|
126 | 415 |
\begin{figure}[t] |
530
b236746e8e4d
futzing with figures (\begin{center|equation} to \centering)
Kevin Walker <kevin@canyon23.net>
parents:
529
diff
changeset
|
416 |
\centering |
508 | 417 |
\begin{tikzpicture} |
418 |
||
419 |
\newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}} |
|
420 |
\newcommand{\nsep}{1.8} |
|
421 |
||
422 |
\node(A) at (0,0) { |
|
423 |
\begin{tikzpicture} |
|
424 |
||
425 |
\draw (0,0) coordinate (p1); |
|
426 |
\draw (3.6,0) coordinate (p2); |
|
427 |
\draw (2.3,1) coordinate (p3); |
|
428 |
\draw (2.3,-1) coordinate (p4); |
|
429 |
||
430 |
\begin{scope} |
|
431 |
\clip (p1) .. controls +(.5,-.5) and +(-.8,0) .. (p4) -- |
|
432 |
(p2) -- (p3) .. controls +(-.8,0) and +(.5,.5) .. (p1); |
|
433 |
\foreach \x in {0,0.26,...,4} { |
|
434 |
\draw[green!50!brown] (\x,0) -- (intersection cs: first line={(p2)--(p3)}, second line={(0,0)--(0,1)}); |
|
435 |
\draw[green!50!brown] (\x,0) -- (intersection cs: first line={(p2)--(p4)}, second line={(0,0)--(0,1)}); |
|
436 |
} |
|
437 |
\end{scope} |
|
438 |
||
439 |
\draw (p1) .. controls ($(p1) + (.5,.5)$) and ($(p3) + (-.8,0)$) .. node[red, above=3pt] {$a$} (p3); |
|
440 |
\draw (p1) .. controls ($(p1) + (.5,-.5)$) and ($(p4) + (-.8,0)$) .. node[red, below=3pt] {$a$} (p4); |
|
441 |
\draw (p3) -- node[red, above=7pt, right=1pt] {$x \times I$} (p2); |
|
442 |
\draw (p4) -- node[red, below=7pt, right=1pt] {$x \times I$} (p2); |
|
443 |
\draw (p1) -- (p2); |
|
444 |
||
445 |
\draw (p1) \vertex; |
|
446 |
\draw (p2) \vertex; |
|
447 |
\draw (p3) \vertex; |
|
448 |
\draw (p4) \vertex; |
|
449 |
||
450 |
\end{tikzpicture} |
|
451 |
}; |
|
452 |
||
453 |
\node[outer sep=\nsep](B) at (5.5,0) { |
|
454 |
\begin{tikzpicture} |
|
455 |
||
456 |
\draw (0,0) coordinate (p1); |
|
457 |
\draw (3.6,0) coordinate (p2); |
|
458 |
\draw (2.3,1) coordinate (p3); |
|
459 |
\draw (2.3,-1) coordinate (p4); |
|
460 |
\draw (4.6,0) coordinate (p2b); |
|
461 |
||
462 |
\begin{scope} |
|
463 |
\clip (p1) .. controls +(.5,-.5) and +(-.8,0) .. (p4) -- |
|
464 |
(p2) -- (p3) .. controls +(-.8,0) and +(.5,.5) .. (p1); |
|
465 |
\foreach \x in {0,0.26,...,4} { |
|
466 |
\draw[green!50!brown] (\x,0) -- (intersection cs: first line={(p2)--(p3)}, second line={(0,0)--(0,1)}); |
|
467 |
\draw[green!50!brown] (\x,0) -- (intersection cs: first line={(p2)--(p4)}, second line={(0,0)--(0,1)}); |
|
468 |
} |
|
469 |
\end{scope} |
|
470 |
||
471 |
\begin{scope} |
|
472 |
\clip (p3)--(p2)--(p4)--(p2b)--cycle; |
|
473 |
\draw[blue!50!brown, step=.23] ($(p4)+(0,-1)$) grid +(3,3); |
|
474 |
\end{scope} |
|
475 |
||
476 |
\draw (p1) .. controls ($(p1) + (.5,.5)$) and ($(p3) + (-.8,0)$) .. node[red, above=3pt] {$a$} (p3); |
|
477 |
\draw (p1) .. controls ($(p1) + (.5,-.5)$) and ($(p4) + (-.8,0)$) .. node[red, below=3pt] {$a$} (p4); |
|
478 |
\draw (p3) -- (p2); |
|
479 |
\draw (p4) -- (p2); |
|
480 |
\draw (p3) -- node[red, above=7pt, right=1pt] {$x \times I$} (p2b); |
|
481 |
\draw (p4) -- node[red, below=7pt, right=1pt] {$x \times I$} (p2b); |
|
482 |
||
483 |
\draw (p1) \vertex; |
|
484 |
\draw (p2) \vertex; |
|
485 |
\draw (p3) \vertex; |
|
486 |
\draw (p4) \vertex; |
|
487 |
\draw (p2b) \vertex; |
|
488 |
||
489 |
\end{tikzpicture} |
|
490 |
}; |
|
491 |
||
492 |
\node[outer sep=\nsep](C) at (11,0) { |
|
493 |
\begin{tikzpicture} |
|
494 |
||
495 |
\draw (0,0) coordinate (p1); |
|
496 |
\draw (2.3,0) coordinate (p2); |
|
497 |
\draw (2.3,1) coordinate (p3); |
|
498 |
\draw (2.3,-1) coordinate (p4); |
|
499 |
\draw (3.6,0) coordinate (p2b); |
|
500 |
||
501 |
\begin{scope} |
|
502 |
\clip (p1) .. controls +(.5,-.5) and +(-.8,0) .. (p4) -- |
|
503 |
(p2) -- (p3) .. controls +(-.8,0) and +(.5,.5) .. (p1); |
|
504 |
\foreach \x in {0,0.26,...,4} { |
|
505 |
\draw[green!50!brown] (\x,-1) -- (\x,1); |
|
506 |
} |
|
507 |
\end{scope} |
|
508 |
||
509 |
\begin{scope} |
|
510 |
\clip (p3)--(p2)--(p4)--(p2b)--cycle; |
|
511 |
\draw[blue!50!brown, step=.23] ($(p4)+(0,-1)$) grid +(3,3); |
|
512 |
\end{scope} |
|
513 |
||
514 |
\draw (p1) .. controls ($(p1) + (.5,.5)$) and ($(p3) + (-.8,0)$) .. node[red, above=3pt] {$a$} (p3); |
|
515 |
\draw (p1) .. controls ($(p1) + (.5,-.5)$) and ($(p4) + (-.8,0)$) .. node[red, below=3pt] {$a$} (p4); |
|
516 |
\draw[green!50!brown] (p3) -- (p4); |
|
517 |
\draw (p3) -- node[red, above=7pt, right=1pt] {$x \times I$} (p2b); |
|
518 |
\draw (p4) -- node[red, below=7pt, right=1pt] {$x \times I$} (p2b); |
|
519 |
||
520 |
\draw (p1) \vertex; |
|
521 |
\draw (p3) \vertex; |
|
522 |
\draw (p4) \vertex; |
|
523 |
\draw (p2b) \vertex; |
|
524 |
||
525 |
\end{tikzpicture} |
|
526 |
}; |
|
527 |
||
528 |
\draw[->, thick, blue!50!green] (A) -- (B); |
|
529 |
\draw[->, thick, blue!50!green] (B) -- node[black, above] {$=$} (C); |
|
530 |
||
531 |
\end{tikzpicture} |
|
498
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
532 |
\caption{Composition of weak identities, 1} |
126 | 533 |
\label{fzo3} |
534 |
\end{figure} |
|
457
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
535 |
In the first step we have inserted a copy of $(x\times I)\times I$. |
125 | 536 |
Figure \ref{fzo4} shows the other case. |
948 | 537 |
|
538 |
\newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}} |
|
539 |
||
126 | 540 |
\begin{figure}[t] |
530
b236746e8e4d
futzing with figures (\begin{center|equation} to \centering)
Kevin Walker <kevin@canyon23.net>
parents:
529
diff
changeset
|
541 |
\centering |
509 | 542 |
\begin{tikzpicture} |
543 |
||
544 |
\newcommand{\nsep}{1.8} |
|
545 |
||
510 | 546 |
\clip (-4,-1.25)--(12,-1.25)--(16,1.25)--(-1,1.25)--cycle; |
547 |
||
548 |
||
509 | 549 |
\node[outer sep=\nsep](A) at (0,0) { |
550 |
\begin{tikzpicture} |
|
551 |
\draw (0,0) coordinate (p1); |
|
552 |
\draw (4,0) coordinate (p2); |
|
553 |
\draw (2.4,0) coordinate (p2a); |
|
554 |
\draw (2,1.2) coordinate (pu); |
|
555 |
\draw (2,-1.2) coordinate (pd); |
|
556 |
||
557 |
\begin{scope} |
|
558 |
\clip (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
559 |
\foreach \t in {0,.065,...,1} { |
|
560 |
\draw[green!50!brown] ($(p1)!\t!(p2a)$) -- +(90 - \t*90 + \t*6 : 4); |
|
561 |
\draw[green!50!brown] ($(p1)!\t!(p2a)$) -- +(-90 + \t*90 - \t*6 : 4); |
|
562 |
} |
|
563 |
\draw[dashed] ($(p2a) + (-.6,3)$)--(p2a)--($(p2a) + (-.6,-3)$); |
|
564 |
\end{scope} |
|
565 |
||
566 |
\draw (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
567 |
\draw (p1)--(p2); |
|
568 |
||
569 |
\draw (p1) \vertex; |
|
570 |
\draw (p2) \vertex; |
|
571 |
\draw (p2a) \vertex; |
|
572 |
\end{tikzpicture} |
|
573 |
}; |
|
574 |
||
575 |
\node[outer sep=\nsep](B) at (5,0) { |
|
576 |
\begin{tikzpicture} |
|
577 |
\draw (0,0) coordinate (p1); |
|
578 |
\draw (4,0) coordinate (p2); |
|
579 |
\draw (2.4,0) coordinate (p2a); |
|
580 |
\draw (2,1.2) coordinate (pu); |
|
581 |
\draw (2,-1.2) coordinate (pd); |
|
582 |
||
583 |
\begin{scope} |
|
584 |
\clip (-.1,3)--($(p2a) + (-.6,3)$)--(p2a)--($(p2a) + (-.6,-3)$)--(-.1,-3)--cycle; |
|
585 |
||
586 |
\begin{scope} |
|
587 |
\clip (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
588 |
\foreach \t in {0,.065,...,1} { |
|
589 |
\draw[green!50!brown] ($(p1)!\t!(p2a)$) -- +(90 - \t*90 + \t*6 : 4); |
|
590 |
\draw[green!50!brown] ($(p1)!\t!(p2a)$) -- +(-90 + \t*90 - \t*6 : 4); |
|
591 |
} |
|
592 |
\draw ($(p2a) + (-.6,3)$)--(p2a)--($(p2a) + (-.6,-3)$); |
|
593 |
\end{scope} |
|
594 |
||
595 |
||
596 |
\draw (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
597 |
%\draw (p1)--(p2); |
|
598 |
\end{scope} |
|
599 |
||
600 |
\begin{scope} |
|
601 |
\clip (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
602 |
\draw ($(p2a) + (-.6,3)$)--(p2a)--($(p2a) + (-.6,-3)$); |
|
603 |
\end{scope} |
|
604 |
||
605 |
\draw (p1) \vertex; |
|
606 |
\draw (p2a) \vertex; |
|
607 |
\end{tikzpicture} |
|
608 |
}; |
|
609 |
||
610 |
\node[outer sep=\nsep](C) at (9,0) { |
|
611 |
\begin{tikzpicture} |
|
612 |
\draw (0,0) coordinate (p1); |
|
613 |
\draw (4,0) coordinate (p2); |
|
614 |
\draw (2,1.2) coordinate (pu); |
|
615 |
\draw (2,-1.2) coordinate (pd); |
|
616 |
||
617 |
\begin{scope} |
|
618 |
\clip (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
619 |
\foreach \t in {0,.045,...,1} { |
|
620 |
\draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); |
|
621 |
} |
|
622 |
\end{scope} |
|
623 |
||
624 |
\draw (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
625 |
||
626 |
\draw (p1) \vertex; |
|
627 |
\draw (p2) \vertex; |
|
628 |
\end{tikzpicture} |
|
629 |
}; |
|
630 |
||
631 |
\draw[->, thick, blue!50!green] (A) -- (B); |
|
632 |
\draw[->, thick, blue!50!green] ($(B) + (1,0)$) -- node[black, above] {$=$} (C); |
|
633 |
||
634 |
\end{tikzpicture} |
|
498
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
635 |
\caption{Composition of weak identities, 2} |
126 | 636 |
\label{fzo4} |
637 |
\end{figure} |
|
934 | 638 |
We notice that a certain subset of the disk is a product region and remove it. |
124 | 639 |
|
934 | 640 |
Given 2-morphisms $f$ and $g$, we define the horizontal composition $f *_h g$ to be any of the four |
641 |
equal 2-morphisms in Figure \ref{fzo5}. |
|
970 | 642 |
Figure \ref{fig:horizontal-compositions-equal} illustrates part of the proof that these four 2-morphisms are equal. |
934 | 643 |
Similar arguments show that horizontal composition is associative. |
127 | 644 |
\begin{figure}[t] |
941 | 645 |
\begin{align*} |
510 | 646 |
\raisebox{-.9cm}{ |
647 |
\begin{tikzpicture} |
|
648 |
\draw (0,0) .. controls +(1,.8) and +(-1,.8) .. node[above] {$b$} (2.9,0) |
|
649 |
.. controls +(-1,-.8) and +(1,-.8) .. node[below] {$a$} (0,0); |
|
650 |
\draw[->, thick, orange!50!brown] (1.45,-.4)-- node[left, black] {$f$} +(0,.8); |
|
651 |
\end{tikzpicture}} |
|
652 |
\;\;\;*_h\;\; |
|
653 |
\raisebox{-.9cm}{ |
|
654 |
\begin{tikzpicture} |
|
655 |
\draw (0,0) .. controls +(1,.8) and +(-1,.8) .. node[above] {$d$} (2.9,0) |
|
656 |
.. controls +(-1,-.8) and +(1,-.8) .. node[below] {$c$} (0,0); |
|
657 |
\draw[->, thick, orange!50!brown] (1.45,-.4)-- node[left, black] {$g$} +(0,.8); |
|
658 |
\end{tikzpicture}} |
|
941 | 659 |
\;&=\; |
510 | 660 |
\raisebox{-1.9cm}{ |
661 |
\begin{tikzpicture} |
|
662 |
\draw (0,0) coordinate (p1); |
|
663 |
\draw (5.8,0) coordinate (p2); |
|
664 |
\draw (2.9,.3) coordinate (pu); |
|
665 |
\draw (2.9,-.3) coordinate (pd); |
|
666 |
\begin{scope} |
|
667 |
\clip (p1) .. controls +(.6,.3) and +(-.5,0) .. (pu) |
|
668 |
.. controls +(.5,0) and +(-.6,.3) .. (p2) |
|
669 |
.. controls +(-.6,-.3) and +(.5,0) .. (pd) |
|
670 |
.. controls +(-.5,0) and +(.6,-.3) .. (p1); |
|
671 |
\foreach \t in {0,.03,...,1} { |
|
672 |
\draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); |
|
673 |
} |
|
674 |
\end{scope} |
|
675 |
\draw (p1) .. controls +(.6,.3) and +(-.5,0) .. (pu) |
|
676 |
.. controls +(.5,0) and +(-.6,.3) .. (p2) |
|
677 |
.. controls +(-.6,-.3) and +(.5,0) .. (pd) |
|
678 |
.. controls +(-.5,0) and +(.6,-.3) .. (p1); |
|
679 |
\draw (p1) .. controls +(1,-2) and +(-1,-1) .. (pd); |
|
680 |
\draw (p2) .. controls +(-1,2) and +(1,1) .. (pu); |
|
681 |
\draw[->, thick, orange!50!brown] (1.45,-1.1)-- node[left, black] {$f$} +(0,.7); |
|
682 |
\draw[->, thick, orange!50!brown] (4.35,.4)-- node[left, black] {$g$} +(0,.7); |
|
683 |
\draw[->, thick, blue!75!yellow] (1.5,.78) node[black, above] {$(b\cdot c)\times I$} -- (2.5,0); |
|
967
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
684 |
\end{tikzpicture}} \displaybreak[1] \\ |
941 | 685 |
\;&=\; |
686 |
\raisebox{-2.1cm}{ |
|
687 |
\begin{tikzpicture} |
|
688 |
\draw (0,0) coordinate (p1); |
|
689 |
\draw (5.8,0) coordinate (p2); |
|
690 |
\draw (2.9,0) coordinate (pu); |
|
691 |
\draw (2.9,-.9) coordinate (pd); |
|
692 |
\begin{scope} |
|
693 |
\clip (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu) |
|
694 |
.. controls +(.5,-.3) and +(-.6,-.3) .. (p2) |
|
695 |
.. controls +(-.6,-.9) and +(.5,0) .. (pd) |
|
696 |
.. controls +(-.5,0) and +(.6,-.9) .. (p1); |
|
697 |
\foreach \t in {0,.03,...,1} { |
|
698 |
\draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); |
|
699 |
} |
|
700 |
\end{scope} |
|
701 |
\draw (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu) |
|
702 |
.. controls +(.5,-.3) and +(-.6,-.3) .. (p2) |
|
703 |
.. controls +(-.6,-.9) and +(.5,0) .. (pd) |
|
704 |
.. controls +(-.5,0) and +(.6,-.9) .. (p1); |
|
705 |
\draw (p1) .. controls +(1,1) and +(-1,1) .. (pu); |
|
706 |
\draw (p2) .. controls +(-1,1) and +(1,1) .. (pu); |
|
707 |
\draw[->, thick, orange!50!brown] (1.45,-0.1)-- node[left, black] {$f$} +(0,.7); |
|
708 |
\draw[->, thick, orange!50!brown] (4.35,-0.1)-- node[left, black] {$g$} +(0,.7); |
|
709 |
\draw[->, thick, blue!75!yellow] (4.3,-1.5) node[black, below] {$(a\cdot c)\times I$} -- (3.3,-0.5); |
|
967
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
710 |
\end{tikzpicture}} \displaybreak[1] \\ |
941 | 711 |
\;&=\; |
712 |
\raisebox{-1.9cm}{ |
|
713 |
\begin{tikzpicture}[y=-1cm] |
|
714 |
\draw (0,0) coordinate (p1); |
|
715 |
\draw (5.8,0) coordinate (p2); |
|
716 |
\draw (2.9,.3) coordinate (pu); |
|
717 |
\draw (2.9,-.3) coordinate (pd); |
|
718 |
\begin{scope} |
|
719 |
\clip (p1) .. controls +(.6,.3) and +(-.5,0) .. (pu) |
|
720 |
.. controls +(.5,0) and +(-.6,.3) .. (p2) |
|
721 |
.. controls +(-.6,-.3) and +(.5,0) .. (pd) |
|
722 |
.. controls +(-.5,0) and +(.6,-.3) .. (p1); |
|
723 |
\foreach \t in {0,.03,...,1} { |
|
724 |
\draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); |
|
725 |
} |
|
726 |
\end{scope} |
|
727 |
\draw (p1) .. controls +(.6,.3) and +(-.5,0) .. (pu) |
|
728 |
.. controls +(.5,0) and +(-.6,.3) .. (p2) |
|
729 |
.. controls +(-.6,-.3) and +(.5,0) .. (pd) |
|
730 |
.. controls +(-.5,0) and +(.6,-.3) .. (p1); |
|
731 |
\draw (p1) .. controls +(1,-2) and +(-1,-1) .. (pd); |
|
732 |
\draw (p2) .. controls +(-1,2) and +(1,1) .. (pu); |
|
733 |
\draw[<-, thick, orange!50!brown] (1.45,-1.1)-- node[left, black] {$f$} +(0,.7); |
|
734 |
\draw[<-, thick, orange!50!brown] (4.35,.4)-- node[left, black] {$g$} +(0,.7); |
|
735 |
\draw[->, thick, blue!75!yellow] (1.5,.78) node[black, below] {$(a\cdot d)\times I$} -- (2.5,0); |
|
967
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
736 |
\end{tikzpicture}} \displaybreak[1] \\ |
941 | 737 |
\;&=\; |
738 |
\raisebox{-1.0cm}{ |
|
739 |
\begin{tikzpicture}[y=-1cm] |
|
740 |
\draw (0,0) coordinate (p1); |
|
741 |
\draw (5.8,0) coordinate (p2); |
|
742 |
\draw (2.9,0) coordinate (pu); |
|
743 |
\draw (2.9,-.9) coordinate (pd); |
|
744 |
\begin{scope} |
|
745 |
\clip (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu) |
|
746 |
.. controls +(.5,-.3) and +(-.6,-.3) .. (p2) |
|
747 |
.. controls +(-.6,-.9) and +(.5,0) .. (pd) |
|
748 |
.. controls +(-.5,0) and +(.6,-.9) .. (p1); |
|
749 |
\foreach \t in {0,.03,...,1} { |
|
750 |
\draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); |
|
751 |
} |
|
752 |
\end{scope} |
|
753 |
\draw (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu) |
|
754 |
.. controls +(.5,-.3) and +(-.6,-.3) .. (p2) |
|
755 |
.. controls +(-.6,-.9) and +(.5,0) .. (pd) |
|
756 |
.. controls +(-.5,0) and +(.6,-.9) .. (p1); |
|
757 |
\draw (p1) .. controls +(1,1) and +(-1,1) .. (pu); |
|
758 |
\draw (p2) .. controls +(-1,1) and +(1,1) .. (pu); |
|
759 |
\draw[<-, thick, orange!50!brown] (1.45,-0.1)-- node[left, black] {$f$} +(0,.7); |
|
760 |
\draw[<-, thick, orange!50!brown] (4.35,-0.1)-- node[left, black] {$g$} +(0,.7); |
|
761 |
\draw[->, thick, blue!75!yellow] (4.3,-1.5) node[black, above] {$(b\cdot d)\times I$} -- (3.3,-0.5); |
|
762 |
\end{tikzpicture}} |
|
763 |
\end{align*} |
|
127 | 764 |
\caption{Horizontal composition of 2-morphisms} |
765 |
\label{fzo5} |
|
766 |
\end{figure} |
|
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
767 |
\begin{figure}[t] |
969
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
768 |
%$$ |
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
769 |
%\mathfig{0.6}{triangle/triangle3c} |
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
770 |
%$$ |
955 | 771 |
$$ |
772 |
\begin{tikzpicture} |
|
773 |
\node (fg1) at (0,0) { |
|
774 |
\begin{tikzpicture}[baseline=-0.6cm] |
|
775 |
\path (0,0) coordinate (f1); |
|
776 |
\path (3,0) coordinate (f2); |
|
777 |
\path (3,-0.5) coordinate (g1); |
|
778 |
\path (6,-0.5) coordinate (g2); |
|
779 |
\node at (1.5,0.125) {$f$}; |
|
780 |
\node at (4.5,-0.625) {$g$}; |
|
781 |
\draw (f1) .. controls +(1,.8) and +(-1,.8) .. (f2); |
|
782 |
\draw[dashed] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2); |
|
783 |
\draw (f1) .. controls +(1,-1) and +(-1,-.4) .. (g1); |
|
784 |
\draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2); |
|
785 |
\draw[dashed] (g1) .. controls +(1,.4) and +(-1,.4) .. (g2); |
|
786 |
\draw (f2) .. controls +(1,.4) and +(-1,1) .. (g2); |
|
787 |
% |
|
969
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
788 |
\draw[blue,->] (-0.8,-1.2) node[below] {$(a \bullet d) \times I$} -- (1,-0.5) ; |
955 | 789 |
\path[clip] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2) |
790 |
.. controls +(1,.4) and +(-1,1) .. (g2) |
|
791 |
.. controls +(-1,.4) and +(1,.4) .. (g1) |
|
792 |
.. controls +(-1,-.4) and +(1,-1) .. (f1); |
|
972
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
793 |
\foreach \x in {0,0.2, ..., 6} { |
955 | 794 |
\draw[green!50!brown] (\x,-2) -- + (0,4); |
795 |
} |
|
796 |
\end{tikzpicture} |
|
797 |
}; |
|
798 |
\node (fg2) at (4,-4) { |
|
799 |
\begin{tikzpicture}[baseline=-0.1cm] |
|
800 |
\path (0,0) coordinate (f1); |
|
801 |
\path (3,0) coordinate (f2); |
|
802 |
\path (3,-0.5) coordinate (g1); |
|
803 |
\path (6,-0.5) coordinate (g2); |
|
804 |
\node at (1.5,0.125) {$f$}; |
|
805 |
\node at (4.5,-0.625) {$g$}; |
|
806 |
\draw[dashed] (f1) .. controls +(1,.8) and +(-1,.8) .. (f2); |
|
807 |
\draw[dashed] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2); |
|
808 |
\draw (f1) .. controls +(1,-1) and +(-1,-.4) .. (g1); |
|
809 |
\draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2); |
|
810 |
\draw[dashed] (g1) .. controls +(1,.4) and +(-1,.4) .. (g2); |
|
811 |
\draw[dashed] (f2) .. controls +(1,.4) and +(-1,1) .. (g2); |
|
812 |
\draw (f1) .. controls +(1,1.5) and +(-1,2)..(g2); |
|
813 |
% |
|
814 |
\begin{scope} |
|
815 |
\path[clip] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2) |
|
816 |
.. controls +(1,.4) and +(-1,1) .. (g2) |
|
817 |
.. controls +(-1,.4) and +(1,.4) .. (g1) |
|
818 |
.. controls +(-1,-.4) and +(1,-1) .. (f1); |
|
972
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
819 |
\foreach \x in {0,0.2, ..., 6} { |
955 | 820 |
\draw[green!50!brown] (\x,-2) -- + (0,4); |
821 |
} |
|
822 |
\end{scope} |
|
823 |
\begin{scope} |
|
824 |
\path[clip] (f1) .. controls +(1,1.5) and +(-1,2).. (g2) |
|
825 |
.. controls +(-1,1) and +(1,.4) .. (f2) |
|
826 |
.. controls +(-1,.8) and + (1,.8) .. (f1); |
|
972
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
827 |
\foreach \x in {0,0.2, ..., 6} { |
955 | 828 |
\draw[green!50!brown] (\x,-2) -- + (0,4); |
829 |
} |
|
830 |
\end{scope} |
|
831 |
\end{tikzpicture} |
|
832 |
}; |
|
833 |
\node (fg3) at (8,0) { |
|
834 |
\begin{tikzpicture}[baseline=-2.45cm] |
|
835 |
\path (0,0) coordinate (f1); |
|
836 |
\path (3,0) coordinate (f2); |
|
837 |
\path (3,0) coordinate (g1); |
|
838 |
\path (6,0) coordinate (g2); |
|
839 |
\node at (1.5,0) {$f$}; |
|
840 |
\node at (4.5,0) {$g$}; |
|
841 |
\draw[dashed] (f1) .. controls +(1,.8) and +(-1,.8) .. (f2); |
|
842 |
\draw (f1) .. controls +(1,-.8) and +(-1,-.8) .. (f2); |
|
843 |
\draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2); |
|
844 |
\draw[dashed] (g1) .. controls +(1,.8) and +(-1,.8) .. (g2); |
|
845 |
\draw (f1) .. controls +(1,1.5) and +(-1,1.5)..(g2); |
|
846 |
% |
|
969
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
847 |
\draw[blue,->] (4,1.75) node[above] {$(b \bullet d) \times I$}-- + (0,-1); |
955 | 848 |
\begin{scope} |
849 |
\path[clip] (f1) .. controls +(1,1.5) and +(-1,1.5).. (g2) |
|
850 |
.. controls +(-1,.8) and +(1,.8) .. (f2) |
|
851 |
.. controls +(-1,.8) and + (1,.8) .. (f1); |
|
972
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
852 |
\foreach \x in {0,0.2, ..., 6} { |
955 | 853 |
\draw[green!50!brown] (\x,-2) -- + (0,4); |
854 |
} |
|
855 |
\end{scope} |
|
856 |
\end{tikzpicture} |
|
857 |
}; |
|
969
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
858 |
\draw[->] ($(fg1.south)+(0,0.5)$) -- node[left=0.5cm] {add $(b \bullet d) \times I$} (fg2); |
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
859 |
\draw[->] (fg2) -- node[right=0.5cm] {remove $(a \bullet d) \times I$} ($(fg3.south)+(0,1.75)$); |
955 | 860 |
\path (fg1) -- node {$=$} (fg3); |
861 |
\end{tikzpicture} |
|
862 |
$$ |
|
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
863 |
\caption{Part of the proof that the four different horizontal compositions of 2-morphisms are equal.} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
864 |
\label{fig:horizontal-compositions-equal} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
865 |
\end{figure} |
125 | 866 |
|
934 | 867 |
Given 1-morphisms $a$, $b$ and $c$ of $D$, we define the associator from $(a\bullet b)\bullet c$ to $a\bullet(b\bullet c)$ |
946 | 868 |
as in Figure \ref{fig:associator}. |
934 | 869 |
This is just a reparameterization of the pinched product $(a\bullet b\bullet c)\times I$ of $\cC$. |
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
870 |
\begin{figure}[t] |
969
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
871 |
%$$ |
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
872 |
%\mathfig{0.4}{triangle/triangle4a} |
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
873 |
%$$ |
948 | 874 |
$$ |
875 |
\begin{tikzpicture} |
|
876 |
\node[circle,fill=black,inner sep=1pt] at (1.73,0) {}; |
|
877 |
\node[circle,fill=black,inner sep=1pt] at (-1.73,0) {}; |
|
878 |
\begin{scope}[yshift=-1cm] |
|
879 |
\path[clip] (0,0) circle (2); |
|
880 |
\begin{scope}[yshift=2cm] |
|
881 |
\draw (0,0) circle (2); |
|
882 |
\node[circle,fill=black,inner sep=1pt] (L2) at (-90:2) {}; |
|
883 |
\node[circle,fill=black,inner sep=1pt] (L1) at (-120:2) {}; |
|
884 |
\node at (-60:2.25) {$c$}; |
|
885 |
\node at (-105:2.25) {$b$}; |
|
886 |
\node at (-135:2.25) {$a$}; |
|
887 |
\end{scope} |
|
888 |
\end{scope} |
|
889 |
\begin{scope}[yshift=1cm] |
|
890 |
\path[clip] (0,0) circle (2); |
|
891 |
\begin{scope}[yshift=-2cm] |
|
892 |
\node at (120:2.25) {$a$}; |
|
893 |
\node at (75:2.25) {$b$}; |
|
894 |
\node at (45:2.25) {$c$}; |
|
895 |
\draw (0,0) circle (2); |
|
896 |
\node[circle,fill=black,inner sep=1pt] (U1) at (90:2) {}; |
|
897 |
\node[circle,fill=black,inner sep=1pt] (U2) at (60:2) {}; |
|
898 |
\end{scope} |
|
899 |
\end{scope} |
|
900 |
\draw[dashed] (L1) -- (U1); |
|
901 |
\draw[dashed] (L2) -- (U2); |
|
902 |
\begin{scope} |
|
903 |
\path[clip] (0,1) circle (2); |
|
904 |
\path[clip] (0,-1) circle (2); |
|
965
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
905 |
\foreach \t in {-2.5,-2.3,...,-1.2} { |
948 | 906 |
\draw[green!50!brown] (\t,-1) -- +(1.19,2); |
907 |
} |
|
965
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
908 |
\foreach \t in {-1.1,-0.9,...,0} { |
948 | 909 |
\draw[blue!50!brown] (\t,-1) -- +(1.19,2); |
910 |
} |
|
965
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
911 |
\foreach \t in {0.1,0.3,...,2.5} { |
948 | 912 |
\draw[red!50!brown] (\t,-1) -- +(1.19,2); |
913 |
} |
|
914 |
\end{scope} |
|
915 |
\draw[->, thick, blue!75!yellow] (-2,0.5) node[black, left] {$a\times I$} -- (-1,0.5); |
|
916 |
\draw[->, thick, blue!75!yellow] (-0.25,-1.5) node[black, below] {$b\times I$} -- (-0.25,-0.5); |
|
917 |
\draw[->, thick, blue!75!yellow] (2,-0.5) node[black, right] {$c\times I$} -- (1,-0.5); |
|
918 |
\end{tikzpicture} |
|
919 |
$$ |
|
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
920 |
\caption{An associator.} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
921 |
\label{fig:associator} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
922 |
\end{figure} |
934 | 923 |
|
935
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
924 |
Let $x,y,z$ be objects of $D$ and let $a:x\to y$ and $b:y\to z$ be 1-morphisms of $D$. |
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
925 |
We have already defined above |
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
926 |
structure maps $u:a\bullet \id_y\to a$ and $v:\id_y\bullet b\to b$, as well as an associator |
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
927 |
$\alpha: (a\bullet \id_y)\bullet b\to a\bullet(\id_y\bullet b)$, as shown in |
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
928 |
Figure \ref{fig:ingredients-triangle-axiom}. |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
929 |
(See also Figures \ref{fzo2} and \ref{fig:associator}.) |
935
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
930 |
We now show that $D$ satisfies the triangle axiom, which states that $u\bullet\id_b$ |
970 | 931 |
is equal to the vertical composition of $\alpha$ and $\id_a\bullet v$. |
935
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
932 |
(Both are 2-morphisms from $(a\bullet \id_y)\bullet b$ to $a\bullet b$.) |
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
933 |
\begin{figure}[t] |
969
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
934 |
%\begin{align*} |
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
935 |
%\mathfig{0.4}{triangle/triangle4a} \\ |
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
936 |
%\mathfig{0.4}{triangle/triangle4b} \\ |
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
937 |
%\mathfig{0.4}{triangle/triangle4c} |
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
938 |
%\end{align*} |
967
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
939 |
\begin{align*} |
972
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
940 |
u & = |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
941 |
\begin{tikzpicture}[baseline] |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
942 |
\coordinate (P) at (0,0); |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
943 |
\coordinate (Q) at (3,0); |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
944 |
\coordinate (R) at (1.5,-0.5); |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
945 |
\draw (P) \vertex to[out=40,in=140] node[above] {$a$} (Q) \vertex -- node[below] {\scriptsize $y \times I$} (R) \vertex to[out=-180,in=-40] node[below] {$a$} (P); |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
946 |
\clip (P) to[out=40,in=140] (Q) -- (R) to[out=-180,in=-40] (P); |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
947 |
\foreach \x in {1,...,9} { |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
948 |
\path (P) to[out=40,in=140] node[pos=\x/10] (PQ\x) {} (Q); |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
949 |
\path (P) to[out=-40,in=-180] node[pos=\x/10] (PR\x) {} (R); |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
950 |
\draw[green!50!brown] (PQ\x.center) -- (PR\x.center); |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
951 |
} |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
952 |
\end{tikzpicture} & |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
953 |
v & = |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
954 |
\begin{tikzpicture}[baseline] |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
955 |
\coordinate (P) at (0,0); |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
956 |
\coordinate (Q) at (3,0); |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
957 |
\coordinate (R) at (1.5,-0.5); |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
958 |
\draw (P) \vertex to[out=40,in=140] node[above] {$b$} (Q) \vertex to[out=-140,in=0] node[below] {$b$} (R) -- node[below] {\scriptsize $y \times I$} (P); |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
959 |
\clip (P) to[out=40,in=140] (Q) to[out=-140,in=0] (R) -- (P); |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
960 |
\foreach \x in {1,...,9} { |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
961 |
\path (P) to[out=40,in=140] node[pos=\x/10] (PQ\x) {} (Q); |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
962 |
\path (R) to[out=0,in=-140] node[pos=\x/10] (RQ\x) {} (Q); |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
963 |
\draw[brown] (PQ\x.center) -- (RQ\x.center); |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
964 |
} |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
965 |
\end{tikzpicture} |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
966 |
\end{align*} |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
967 |
\vspace{-2cm} |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
968 |
\begin{align*} |
965
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
969 |
\alpha & = |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
970 |
\begin{tikzpicture}[baseline] |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
971 |
\node[circle,fill=black,inner sep=1pt] at (1.73,0) {}; |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
972 |
\node[circle,fill=black,inner sep=1pt] at (-1.73,0) {}; |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
973 |
\begin{scope}[yshift=-1cm] |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
974 |
\path[clip] (0,0) circle (2); |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
975 |
\begin{scope}[yshift=2cm] |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
976 |
\draw (0,0) circle (2); |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
977 |
\node[circle,fill=black,inner sep=1pt] (L2) at (-90:2) {}; |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
978 |
\node[circle,fill=black,inner sep=1pt] (L1) at (-120:2) {}; |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
979 |
\node at (-60:2.25) {$b$}; |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
980 |
\node at (-105:2.25) {\scriptsize $y \times I$}; |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
981 |
\node at (-135:2.25) {$a$}; |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
982 |
\end{scope} |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
983 |
\end{scope} |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
984 |
\begin{scope}[yshift=1cm] |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
985 |
\path[clip] (0,0) circle (2); |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
986 |
\begin{scope}[yshift=-2cm] |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
987 |
\node at (120:2.25) {$a$}; |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
988 |
\node at (75:2.25) {\scriptsize $y \times I$}; |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
989 |
\node at (45:2.25) {$b$}; |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
990 |
\draw (0,0) circle (2); |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
991 |
\node[circle,fill=black,inner sep=1pt] (U1) at (90:2) {}; |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
992 |
\node[circle,fill=black,inner sep=1pt] (U2) at (60:2) {}; |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
993 |
\end{scope} |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
994 |
\end{scope} |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
995 |
\draw[dashed] (L1) -- (U1); |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
996 |
\draw[dashed] (L2) -- (U2); |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
997 |
\begin{scope} |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
998 |
\path[clip] (0,1) circle (2); |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
999 |
\path[clip] (0,-1) circle (2); |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
1000 |
\foreach \t in {-2.5,-2.3,...,-1.2} { |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
1001 |
\draw[green!50!brown] (\t,-1) -- +(1.19,2); |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
1002 |
} |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
1003 |
\foreach \t in {-1.1,-0.9,...,0} { |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
1004 |
\draw[blue!50!brown] (\t,-1) -- +(1.19,2); |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
1005 |
} |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
1006 |
\foreach \t in {0.1,0.3,...,2.5} { |
972
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
1007 |
\draw[brown] (\t,-1) -- +(1.19,2); |
965
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
1008 |
} |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
1009 |
\foreach \x in {0,1,...,10} { |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
1010 |
\path (L1) to[out=60,in=-120] node[pos=\x/10] (p1\x) {}(U1); |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
1011 |
\path (L2) to[out=60,in=-120] node[pos=\x/10] (p2\x) {} (U2); |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
1012 |
\draw[blue!50!brown] (p1\x.center) -- (p2\x.center); |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
1013 |
} |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
1014 |
\end{scope} |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
1015 |
\draw[->, thick, blue!75!yellow] (-2,0.5) node[black, left] {$a\times I$} -- (-1,0.5); |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
1016 |
\draw[->, thick, blue!75!yellow] (-0.1,-1.5) node[black, below] {$y\times I \times I$} -- (-0.25,-0.5); |
987af9e53631
more diagrams, doing the complicated one in tikz again now that I've discovered pos (and an annoying bug in tikz)
Scott Morrison <scott@tqft.net>
parents:
963
diff
changeset
|
1017 |
\draw[->, thick, blue!75!yellow] (2,-0.5) node[black, right] {$b\times I$} -- (1,-0.5); |
972
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
1018 |
\end{tikzpicture} |
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1019 |
\end{align*} |
972
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
1020 |
\vspace{-1cm} |
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1021 |
\caption{Ingredients for the triangle axiom.} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1022 |
\label{fig:ingredients-triangle-axiom} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1023 |
\end{figure} |
935
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
1024 |
|
970 | 1025 |
The horizontal compositions $u *_h \id_b$ and $\id_a *_h v$ are shown in Figure \ref{fig:horizontal-composition} |
935
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
1026 |
(see also Figure \ref{fzo5}). |
970 | 1027 |
The vertical composition of $\alpha$ and $\id_a *_h v$ is shown in Figure \ref{fig:vertical-composition}. |
1028 |
Figure \ref{fig:adding-a-collar} shows that we can add a collar to $u *_h \id_b$ so that the result differs from |
|
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1029 |
Figure \ref{fig:vertical-composition} by an isotopy rel boundary. |
935
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
1030 |
Note that here we have used in an essential way the associativity of product morphisms (Axiom \ref{axiom:product}.3) |
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
1031 |
as well as compatibility of product morphisms with fiber-preserving maps (Axiom \ref{axiom:product}.1). |
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1032 |
\begin{figure}[t] |
969
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
1033 |
%\begin{align*} |
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
1034 |
%\mathfig{0.4}{triangle/triangle4d} |
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
1035 |
%\mathfig{0.4}{triangle/triangle4e} \\ |
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
1036 |
%\end{align*} |
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1037 |
\begin{align*} |
969
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
1038 |
u *_h (b \times I) & = |
967
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1039 |
\begin{tikzpicture}[baseline] |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1040 |
\coordinate (L) at (0,0); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1041 |
\coordinate (R) at (3,0); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1042 |
\coordinate (T) at (1.5,0.8); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1043 |
\coordinate (M) at (1.5,-0.4); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1044 |
\coordinate (B) at (0.75,-0.8); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1045 |
\path (L) \vertex (T) \vertex (R) \vertex (M) \vertex (B) \vertex; |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1046 |
\draw (L) to[out=40,in=180] node[above] {$a$} (T) |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1047 |
to[out=0,in=140] node[above] {$b$} (R) |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1048 |
to[out=-150,in=-20] node[below] {$b$} (M) |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1049 |
-- node[below] {\tiny $y \times I$} (B) |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1050 |
to[out=150,in=-45] node[below] {$a$} (L); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1051 |
\draw[dashed] (L) to[out=0,in=150] (M) |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1052 |
to[out=30,in=150] (R); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1053 |
\draw[dashed] (T) -- (M); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1054 |
\foreach \n in {0,...,7} { |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1055 |
\path (L) to[out=-45,in=150] node[coordinate,pos=\n/8] (LB\n) {} (B); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1056 |
\path (L) to[out=0,in=150] node[coordinate,pos=\n/8] (LM\n) {} (M); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1057 |
\path (L) to[out=40,in=180] node[coordinate,pos=\n/8] (LT\n) {} (T); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1058 |
\draw[green!50!brown] (LB\n) -- (LM\n) -- (LT\n); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1059 |
} |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1060 |
\foreach \n in {1,...,7} { |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1061 |
\path (M) to[out=-20,in=-150] node[coordinate,pos=\n/8] (MR\n) {} (R); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1062 |
\path (T) to[out=0,in=140] node[coordinate,pos=\n/8] (TR\n) {} (R); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1063 |
\draw[brown] (MR\n) -- (TR\n); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1064 |
} |
972
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
1065 |
\end{tikzpicture} & |
969
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
1066 |
(a \times I) *_h v & = |
967
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1067 |
\begin{tikzpicture}[baseline] |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1068 |
\coordinate (L) at (0,0); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1069 |
\coordinate (R) at (3,0); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1070 |
\coordinate (T) at (1.5,0.8); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1071 |
\coordinate (M) at (1.5,-0.4); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1072 |
\coordinate (B) at (2.25,-0.8); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1073 |
\path (L) \vertex (T) \vertex (R) \vertex (M) \vertex (B) \vertex; |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1074 |
\draw (L) to[out=40,in=180] node[above] {$a$} (T) |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1075 |
to[out=0,in=140] node[above] {$b$} (R) |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1076 |
to[out=-150,in=45] node[below] {$b$} (B) |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1077 |
-- node[below=2] {\tiny $y \times I$} (M) |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1078 |
to[out=-160,in=-30] node[below] {$a$} (L); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1079 |
\draw[dashed] (L) to[out=30,in=150] (M) |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1080 |
to[out=30,in=180] (R); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1081 |
\draw[dashed] (T) -- (M); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1082 |
\foreach \n in {1,...,7} { |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1083 |
\path (B) to[out=45,in=-150] node[coordinate,pos=\n/8] (BR\n) {} (R); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1084 |
\path (M) to[out=30,in=180] node[coordinate,pos=\n/8] (MR\n) {} (R); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1085 |
\path (T) to[out=0,in=140] node[coordinate,pos=\n/8] (TR\n) {} (R); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1086 |
\draw[brown] (BR\n) -- (MR\n) -- (TR\n); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1087 |
} |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1088 |
\foreach \n in {1,...,7} { |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1089 |
\path (L) to[out=-20,in=-150] node[coordinate,pos=\n/8] (LM\n) {} (M); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1090 |
\path (L) to[out=40,in=180] node[coordinate,pos=\n/8] (LT\n) {} (T); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1091 |
\draw[green!50!brown] (LM\n) -- (LT\n); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1092 |
} |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1093 |
\end{tikzpicture} |
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1094 |
\end{align*} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1095 |
\caption{Horizontal compositions in the triangle axiom.} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1096 |
\label{fig:horizontal-composition} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1097 |
\end{figure} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1098 |
\begin{figure}[t] |
972
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
1099 |
\vspace{-1.5cm} |
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1100 |
\begin{align*} |
969
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
1101 |
%\mathfig{0.4}{triangle/triangle4f} \\ |
966
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1102 |
\begin{tikzpicture} |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1103 |
\node[circle,fill=black,inner sep=1pt] (A) at (1.73,0) {}; |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1104 |
\node[circle,fill=black,inner sep=1pt] (B) at (-1.73,0) {}; |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1105 |
\draw[dashed] (A) -- (B); |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1106 |
\node[circle,fill=black,inner sep=1pt] (C) at (0,0) {}; |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1107 |
\node[circle,fill=black,inner sep=1pt] (D) at (0.8,0) {}; |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1108 |
\begin{scope}[yshift=-1cm] |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1109 |
\path[clip] (0,0) circle (2); |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1110 |
\begin{scope}[yshift=2cm] |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1111 |
\draw (0,0) circle (2); |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1112 |
\node[circle,fill=black,inner sep=1pt] (L2) at (-90:2) {}; |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1113 |
\node[circle,fill=black,inner sep=1pt] (L1) at (-120:2) {}; |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1114 |
\end{scope} |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1115 |
\end{scope} |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1116 |
\begin{scope}[yshift=1cm] |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1117 |
\path[clip] (0,0) circle (2); |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1118 |
\begin{scope}[yshift=-2cm] |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1119 |
\draw (0,0) circle (2); |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1120 |
\node[circle,fill=black,inner sep=1pt] (U) at (90:2) {}; |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1121 |
\end{scope} |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1122 |
\end{scope} |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1123 |
\begin{scope} |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1124 |
\path[clip] (0,1) circle (2); |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1125 |
\path[clip] (0,-1) circle (2); |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1126 |
\foreach \n in {1,...,6} { |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1127 |
\path (B) to[out=0,in=-180] node[coordinate,pos=\n/6] (BC\n) {} (C); |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1128 |
\draw[green!50!brown] (BC\n) -- +(0,1); |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1129 |
\draw[green!50!brown] (BC\n) -- +(-1.5,-1); |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1130 |
} |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1131 |
\foreach \n in {0,...,5} { |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1132 |
\path (C) to[out=0,in=-180] node[coordinate,pos=\n/5] (CD\n) {} (D); |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1133 |
\path (L1) to[out=-30,in=-180] node[coordinate,pos=\n/5] (L1L2\n) {} (L2); |
972
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
1134 |
\draw[blue!50!brown] (CD\n) -- (L1L2\n); |
966
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1135 |
} |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1136 |
\foreach \n in {0,...,5} { |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1137 |
\path (L1) to[out=40,in=-140] node[coordinate,pos=\n/5] (L1C\n) {} (C); |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1138 |
\path (L2) to[out=50,in=-130] node[coordinate,pos=\n/5] (L2D\n) {} (D); |
972
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
1139 |
\draw[blue!50!brown] (L1C\n) -- (L2D\n); |
966
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1140 |
} |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1141 |
\foreach \n in {1,3,...,9} { |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1142 |
\path(C) to[out=30,in=150] node[coordinate,pos=\n/10] (CA\n) {} (A); |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1143 |
\path(D) to[out=0,in=180] node[coordinate,pos=\n/10] (DA\n) {} (A); |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1144 |
\path(L2) to[out=0,in=-120] node[coordinate,pos=\n/10] (L2A\n) {} (A); |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1145 |
\draw[brown] (CA\n) -- +(0,1); |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1146 |
\draw[brown] (CA\n) to[out=0] (DA\n) -- (L2A\n); |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1147 |
} |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1148 |
\draw[dashed] (U) -- (C) -- (L1) (L2) -- (D); |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1149 |
\end{scope} |
d9e4d7e1cbf6
redone the complicated figure in tikz
Scott Morrison <scott@tqft.net>
parents:
965
diff
changeset
|
1150 |
\end{tikzpicture} |
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1151 |
\end{align*} |
972
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
1152 |
\vspace{-2.5cm} |
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1153 |
\caption{Vertical composition in the triangle axiom.} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1154 |
\label{fig:vertical-composition} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1155 |
\end{figure} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1156 |
\begin{figure}[t] |
969
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
1157 |
%\begin{align*} |
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
1158 |
%\mathfig{0.4}{triangle/triangle5} |
2232d94b90b8
remoed hand drawn figs; adjusted some composition symbols
Kevin Walker <kevin@canyon23.net>
parents:
967
diff
changeset
|
1159 |
%\end{align*} |
967
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1160 |
\begin{align*} |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1161 |
\begin{tikzpicture}[baseline] |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1162 |
\coordinate (L) at (0,0); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1163 |
\coordinate (R) at (3,0); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1164 |
\coordinate (T) at (1.5,0.8); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1165 |
\coordinate (M) at (1.5,-0.4); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1166 |
\coordinate (B) at (0.75,-0.8); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1167 |
\path (L) \vertex (T) \vertex (R) \vertex (M) \vertex (B) \vertex; |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1168 |
\draw (L) to[out=40,in=180] (T) |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1169 |
to[out=0,in=140] (R) |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1170 |
to[out=-150,in=-20] (M) |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1171 |
-- (B) |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1172 |
to[out=150,in=-45] (L); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1173 |
\draw[dashed] (T) -- (M); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1174 |
\foreach \n in {0,...,7} { |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1175 |
\path (L) to[out=-45,in=150] node[coordinate,pos=\n/8] (LB\n) {} (B); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1176 |
\path (L) to[out=0,in=150] node[coordinate,pos=\n/8] (LM\n) {} (M); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1177 |
\path (L) to[out=40,in=180] node[coordinate,pos=\n/8] (LT\n) {} (T); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1178 |
\draw[green!50!brown] (LB\n) -- (LM\n) -- (LT\n); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1179 |
} |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1180 |
\foreach \n in {1,...,7} { |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1181 |
\path (M) to[out=-20,in=-150] node[coordinate,pos=\n/8] (MR\n) {} (R); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1182 |
\path (T) to[out=0,in=140] node[coordinate,pos=\n/8] (TR\n) {} (R); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1183 |
\draw[brown] (MR\n) -- (TR\n); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1184 |
} |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1185 |
\coordinate (B') at ($(B)+(0.125,-0.25)$); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1186 |
\coordinate (M') at ($(M)+(0.125,-0.25)$); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1187 |
\coordinate (R') at ($(R)+(0.125,-0.25)$); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1188 |
\coordinate (X) at ($(M')+(0,-0.6)$); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1189 |
\path (X) \vertex; |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1190 |
\draw[dashed] (M') -- (X); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1191 |
\draw[clip] (B') to[out=-30,in=180] (X) to[out=0,in=-135] (R') to[out=-150,in=-20] (M') -- cycle; |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1192 |
\foreach \n in {1,...,7} { |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1193 |
\path (M') to[out=-20,in=-150] node[coordinate,pos=\n/8] (M'R'\n) {} (R'); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1194 |
\draw[brown] (M'R'\n) -- +(0,-1); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1195 |
} |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1196 |
\foreach \n in {1,...,5} { |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1197 |
\path (B') to[out=-30,in=180] node[coordinate,pos=\n/6] (B'X\n) {} (X); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1198 |
\path (M') to[out=-90,in=90] node[coordinate,pos=\n/6] (M'X\n) {} (X); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1199 |
\path (B') to[out=30,in=-150] node[coordinate,pos=\n/6] (B'M'\n) {} (M'); |
972
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
1200 |
\draw[blue!50!brown] (B'X\n) -- (M'X\n); |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
1201 |
\draw[blue!50!brown] (B'M'\n) -- (B'X\n); |
967
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1202 |
} |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1203 |
\end{tikzpicture} |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1204 |
\xrightarrow{\text{collar}} |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1205 |
\begin{tikzpicture}[baseline] |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1206 |
\coordinate (L) at (0,0); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1207 |
\coordinate (R) at (3,0); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1208 |
\coordinate (T) at (1.5,0.8); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1209 |
\coordinate (M) at (1.5,-0.4); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1210 |
\coordinate (B) at (0.75,-0.8); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1211 |
\path (L) \vertex (T) \vertex (R) \vertex (M) \vertex (B) \vertex; |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1212 |
\draw (L) to[out=40,in=180] (T) |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1213 |
to[out=0,in=140] (R); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1214 |
\draw[dashed] (R) to[out=-150,in=-20] (M) |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1215 |
-- (B); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1216 |
\draw (B) to[out=150,in=-45] (L); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1217 |
\draw[dashed] (T) -- (M); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1218 |
\foreach \n in {0,...,7} { |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1219 |
\path (L) to[out=-45,in=150] node[coordinate,pos=\n/8] (LB\n) {} (B); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
parents:
966
diff
changeset
|
1220 |
\path (L) to[out=0,in=150] node[coordinate,pos=\n/8] (LM\n) {} (M); |
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
Scott Morrison <scott@tqft.net>
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|
1221 |
\path (L) to[out=40,in=180] node[coordinate,pos=\n/8] (LT\n) {} (T); |
6e4f0ed47e0e
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|
1222 |
\draw[green!50!brown] (LB\n) -- (LM\n) -- (LT\n); |
6e4f0ed47e0e
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|
1223 |
} |
6e4f0ed47e0e
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|
1224 |
\foreach \n in {1,...,7} { |
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|
1225 |
\path (M) to[out=-20,in=-150] node[coordinate,pos=\n/8] (MR\n) {} (R); |
6e4f0ed47e0e
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|
1226 |
\path (T) to[out=0,in=140] node[coordinate,pos=\n/8] (TR\n) {} (R); |
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|
1227 |
\draw[brown] (MR\n) -- (TR\n); |
6e4f0ed47e0e
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|
1228 |
} |
6e4f0ed47e0e
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changeset
|
1229 |
\coordinate (B') at (B); |
6e4f0ed47e0e
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changeset
|
1230 |
\coordinate (M') at (M); |
6e4f0ed47e0e
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changeset
|
1231 |
\coordinate (R') at (R); |
6e4f0ed47e0e
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changeset
|
1232 |
\coordinate (X) at ($(M')+(0,-0.6)$); |
6e4f0ed47e0e
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|
1233 |
\path (X) \vertex; |
6e4f0ed47e0e
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|
1234 |
\draw[dashed] (M') -- (X); |
6e4f0ed47e0e
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|
1235 |
\draw (B') to[out=-30,in=180] (X) to[out=0,in=-135] (R'); |
6e4f0ed47e0e
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|
1236 |
\path[clip] (B') to[out=-30,in=180] (X) to[out=0,in=-135] (R') to[out=-150,in=-20] (M') -- cycle; |
6e4f0ed47e0e
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|
1237 |
\foreach \n in {1,...,7} { |
6e4f0ed47e0e
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|
1238 |
\path (M') to[out=-20,in=-150] node[coordinate,pos=\n/8] (M'R'\n) {} (R'); |
6e4f0ed47e0e
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changeset
|
1239 |
\draw[brown] (M'R'\n) -- +(0,-1); |
6e4f0ed47e0e
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changeset
|
1240 |
} |
6e4f0ed47e0e
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changeset
|
1241 |
\foreach \n in {1,...,5} { |
6e4f0ed47e0e
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|
1242 |
\path (B') to[out=-30,in=180] node[coordinate,pos=\n/6] (B'X\n) {} (X); |
6e4f0ed47e0e
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|
1243 |
\path (M') to[out=-90,in=90] node[coordinate,pos=\n/6] (M'X\n) {} (X); |
6e4f0ed47e0e
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changeset
|
1244 |
\path (B') to[out=30,in=-150] node[coordinate,pos=\n/6] (B'M'\n) {} (M'); |
972
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
1245 |
\draw[blue!50!brown] (B'X\n) -- (M'X\n); |
27c075c68816
final tweaks to figures in C2
Scott Morrison <scott@tqft.net>
parents:
970
diff
changeset
|
1246 |
\draw[blue!50!brown] (B'M'\n) -- (B'X\n); |
967
6e4f0ed47e0e
finished all the figures. (colors need adjusting)
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parents:
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changeset
|
1247 |
} |
6e4f0ed47e0e
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diff
changeset
|
1248 |
\end{tikzpicture} |
6e4f0ed47e0e
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parents:
966
diff
changeset
|
1249 |
\end{align*} |
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1250 |
\caption{Adding a collar in the proof of the triangle axiom.} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1251 |
\label{fig:adding-a-collar} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
1252 |
\end{figure} |
935
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
1253 |
|
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
1254 |
|
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
1255 |
|
512
050dba5e7bdd
fixing some (but not all!?) of the hyperref warnings; start on revision of evmap
Kevin Walker <kevin@canyon23.net>
parents:
510
diff
changeset
|
1256 |
\subsection{\texorpdfstring{$A_\infty$}{A-infinity} 1-categories} |
194 | 1257 |
\label{sec:comparing-A-infty} |
431 | 1258 |
In this section, we make contact between the usual definition of an $A_\infty$ category |
888
a0fd6e620926
Backed out changeset 7abe7642265e
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parents:
865
diff
changeset
|
1259 |
and our definition of a disk-like $A_\infty$ $1$-category, from \S \ref{ss:n-cat-def}. |
477
86c8e2129355
radically shorter a-inf appendix
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diff
changeset
|
1260 |
|
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
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457
diff
changeset
|
1261 |
\medskip |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1262 |
|
888
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
1263 |
Given a disk-like $A_\infty$ $1$-category $\cC$, we define an ``$m_k$-style" |
477
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1264 |
$A_\infty$ $1$-category $A$ as follows. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1265 |
The objects of $A$ are $\cC(pt)$. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1266 |
The morphisms of $A$, from $x$ to $y$, are $\cC(I; x, y)$ |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1267 |
($\cC$ applied to the standard interval with boundary labeled by $x$ and $y$). |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1268 |
For simplicity we will now assume there is only one object and suppress it from the notation. |
887
ab0b4827c89c
more referee report stuff, relatively minor
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
1269 |
Henceforth $A$ will also denote its unique morphism space. |
477
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1270 |
|
854 | 1271 |
A choice of homeomorphism $I\cup I \to I$ induces a chain map $m_2: A\otimes A\to A$. |
477
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1272 |
We now have two different homeomorphisms $I\cup I\cup I \to I$, but they are isotopic. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1273 |
Choose a specific 1-parameter family of homeomorphisms connecting them; this induces |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1274 |
a degree 1 chain homotopy $m_3:A\ot A\ot A\to A$. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1275 |
Proceeding in this way we define the rest of the $m_i$'s. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1276 |
It is straightforward to verify that they satisfy the necessary identities. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1277 |
|
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1278 |
\medskip |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1279 |
|
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1280 |
In the other direction, we start with an alternative conventional definition of an $A_\infty$ algebra: |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1281 |
an algebra $A$ for the $A_\infty$ operad. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1282 |
(For simplicity, we are assuming our $A_\infty$ 1-category has only one object.) |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1283 |
We are free to choose any operad with contractible spaces, so we choose the operad |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1284 |
whose $k$-th space is the space of decompositions of the standard interval $I$ into $k$ |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1285 |
parameterized copies of $I$. |
480 | 1286 |
Note in particular that when $k=1$ this implies a $C_*(\Homeo(I))$ action on $A$. |
1287 |
(Compare with Example \ref{ex:e-n-alg} and the discussion which precedes it.) |
|
477
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1288 |
Given a non-standard interval $J$, we define $\cC(J)$ to be |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1289 |
$(\Homeo(I\to J) \times A)/\Homeo(I\to I)$, |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1290 |
where $\beta \in \Homeo(I\to I)$ acts via $(f, a) \mapsto (f\circ \beta\inv, \beta_*(a))$. |
480 | 1291 |
Note that $\cC(J) \cong A$ (non-canonically) for all intervals $J$. |
477
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1292 |
We define a $\Homeo(J)$ action on $\cC(J)$ via $g_*(f, a) = (g\circ f, a)$. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1293 |
The $C_*(\Homeo(J))$ action is defined similarly. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1294 |
|
887
ab0b4827c89c
more referee report stuff, relatively minor
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
1295 |
Let $J_1$ and $J_2$ be intervals, and let $J_1\cup J_2$ denote their union along a single boundary point. |
477
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1296 |
We must define a map $\cC(J_1)\ot\cC(J_2)\to\cC(J_1\cup J_2)$. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1297 |
Choose a homeomorphism $g:I\to J_1\cup J_2$. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1298 |
Let $(f_i, a_i)\in \cC(J_i)$. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1299 |
We have a parameterized decomposition of $I$ into two intervals given by |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1300 |
$g\inv \circ f_i$, $i=1,2$. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1301 |
Corresponding to this decomposition the operad action gives a map $\mu: A\ot A\to A$. |
86c8e2129355
radically shorter a-inf appendix
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parents:
457
diff
changeset
|
1302 |
Define the gluing map to send $(f_1, a_1)\ot (f_2, a_2)$ to $(g, \mu(a_1\ot a_2))$. |
480 | 1303 |
Operad associativity for $A$ implies that this gluing map is independent of the choice of |
1304 |
$g$ and the choice of representative $(f_i, a_i)$. |
|
477
86c8e2129355
radically shorter a-inf appendix
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457
diff
changeset
|
1305 |
|
888
a0fd6e620926
Backed out changeset 7abe7642265e
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865
diff
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|
1306 |
It is straightforward to verify the remaining axioms for a disk-like $A_\infty$ 1-category. |
477
86c8e2129355
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parents:
457
diff
changeset
|
1307 |
|
86c8e2129355
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457
diff
changeset
|
1308 |
|
86c8e2129355
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457
diff
changeset
|
1309 |
|
86c8e2129355
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457
diff
changeset
|
1310 |
|
86c8e2129355
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diff
changeset
|
1311 |
|
86c8e2129355
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diff
changeset
|
1312 |
|
86c8e2129355
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457
diff
changeset
|
1313 |
|
86c8e2129355
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|
1314 |
\noop { %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
194 | 1315 |
|
431 | 1316 |
That definition associates a chain complex to every interval, and we begin by giving an alternative definition that is entirely in terms of the chain complex associated to the standard interval $[0,1]$. |
194 | 1317 |
\begin{defn} |
345 | 1318 |
A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$, |
1319 |
and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with |
|
194 | 1320 |
\begin{itemize} |
1321 |
\item an action of the operad of $\Obj(\cC)$-labeled cell decompositions |
|
1322 |
\item and a compatible action of $\CD{[0,1]}$. |
|
1323 |
\end{itemize} |
|
1324 |
\end{defn} |
|
345 | 1325 |
Here the operad of cell decompositions of $[0,1]$ has operations indexed by a finite set of |
1326 |
points $0 < x_1< \cdots < x_k < 1$, cutting $[0,1]$ into subintervals. |
|
1327 |
An $X$-labeled cell decomposition labels $\{0, x_1, \ldots, x_k, 1\}$ by $X$. |
|
1328 |
Given two cell decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$, we can compose |
|
1329 |
them to form a new cell decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$ by inserting the points |
|
1330 |
of $\cJ^{(2)}$ linearly into the $m$-th interval of $\cJ^{(1)}$. |
|
1331 |
In the $X$-labeled case, we insist that the appropriate labels match up. |
|
1332 |
Saying we have an action of this operad means that for each labeled cell decomposition |
|
1333 |
$0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain |
|
431 | 1334 |
map $$\cC_{a_0,a_1} \tensor \cdots \tensor \cC_{a_k,a_{k+1}} \to \cC_{a_0,a_{k+1}}$$ and these |
345 | 1335 |
chain maps compose exactly as the cell decompositions. |
1336 |
An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad |
|
1337 |
if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which |
|
1338 |
is supported on the subintervals determined by $\pi$, then the two possible operations |
|
1339 |
(glue intervals together, then apply the diffeomorphisms, or apply the diffeormorphisms |
|
1340 |
separately to the subintervals, then glue) commute (as usual, up to a weakly unique homotopy). |
|
194 | 1341 |
|
345 | 1342 |
Translating between this notion and the usual definition of an $A_\infty$ category is now straightforward. |
1343 |
To restrict to the standard interval, define $\cC_{a,b} = \cC([0,1];a,b)$. |
|
1344 |
Given a cell decomposition $0 < x_1< \cdots < x_k < 1$, we use the map (suppressing labels) |
|
194 | 1345 |
$$\cC([0,1])^{\tensor k+1} \to \cC([0,x_1]) \tensor \cdots \tensor \cC[x_k,1] \to \cC([0,1])$$ |
345 | 1346 |
where the factors of the first map are induced by the linear isometries $[0,1] \to [x_i, x_{i+1}]$, and the second map is just gluing. |
1347 |
The action of $\CD{[0,1]}$ carries across, and is automatically compatible. |
|
1348 |
Going the other way, we just declare $\cC(J;a,b) = \cC_{a,b}$, pick a diffeomorphism |
|
1349 |
$\phi_J : J \isoto [0,1]$ for every interval $J$, define the gluing map |
|
1350 |
$\cC(J_1) \tensor \cC(J_2) \to \cC(J_1 \cup J_2)$ by the first applying |
|
1351 |
the cell decomposition map for $0 < \frac{1}{2} < 1$, then the self-diffeomorphism of $[0,1]$ |
|
1352 |
given by $\frac{1}{2} (\phi_{J_1} \cup (1+ \phi_{J_2})) \circ \phi_{J_1 \cup J_2}^{-1}$. |
|
1353 |
You can readily check that this gluing map is associative on the nose. \todo{really?} |
|
194 | 1354 |
|
1355 |
%First recall the \emph{coloured little intervals operad}. Given a set of labels $\cL$, the operations are indexed by \emph{decompositions of the interval}, each of which is a collection of disjoint subintervals $\{(a_i,b_i)\}_{i=1}^k$ of $[0,1]$, along with a labeling of the complementary regions by $\cL$, $\{l_0, \ldots, l_k\}$. Given two decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$ such that $l^{(1)}_{m-1} = l^{(2)}_0$ and $l^{(1)}_{m} = l^{(2)}_{k^{(2)}}$, we can form a new decomposition by inserting the intervals of $\cJ^{(2)}$ linearly inside the $m$-th interval of $\cJ^{(1)}$. We call the resulting decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$. |
|
1356 |
||
1357 |
%\begin{defn} |
|
1358 |
%A \emph{topological $A_\infty$ category} $\cC$ has a set of objects $\Obj(\cC)$ and for each $a,b \in \Obj(\cC)$ a chain complex $\cC_{a,b}$, along with a compatible `composition map' and an `action of families of diffeomorphisms'. |
|
1359 |
||
1360 |
%A \emph{composition map} $f$ is a family of chain maps, one for each decomposition of the interval, $f_\cJ : A^{\tensor k} \to A$, making $\cC$ into a category over the coloured little intervals operad, with labels $\cL = \Obj(\cC)$. Thus the chain maps satisfy the identity |
|
1361 |
%\begin{equation*} |
|
1362 |
%f_{\cJ^{(1)} \circ_m \cJ^{(2)}} = f_{\cJ^{(1)}} \circ (\id^{\tensor m-1} \tensor f_{\cJ^{(2)}} \tensor \id^{\tensor k^{(1)} - m}). |
|
1363 |
%\end{equation*} |
|
1364 |
||
1365 |
%An \emph{action of families of diffeomorphisms} is a chain map $ev: \CD{[0,1]} \tensor A \to A$, such that |
|
1366 |
%\begin{enumerate} |
|
1367 |
%\item The diagram |
|
1368 |
%\begin{equation*} |
|
1369 |
%\xymatrix{ |
|
1370 |
%\CD{[0,1]} \tensor \CD{[0,1]} \tensor A \ar[r]^{\id \tensor ev} \ar[d]^{\circ \tensor \id} & \CD{[0,1]} \tensor A \ar[d]^{ev} \\ |
|
1371 |
%\CD{[0,1]} \tensor A \ar[r]^{ev} & A |
|
1372 |
%} |
|
1373 |
%\end{equation*} |
|
1374 |
%commutes up to weakly unique homotopy. |
|
1375 |
%\item If $\phi \in \Diff([0,1])$ and $\cJ$ is a decomposition of the interval, we obtain a new decomposition $\phi(\cJ)$ and a collection $\phi_m \in \Diff([0,1])$ of diffeomorphisms obtained by taking the restrictions $\restrict{\phi}{[a_m,b_m]} : [a_m,b_m] \to [\phi(a_m),\phi(b_m)]$ and pre- and post-composing these with the linear diffeomorphisms $[0,1] \to [a_m,b_m]$ and $[\phi(a_m),\phi(b_m)] \to [0,1]$. We require that |
|
1376 |
%\begin{equation*} |
|
1377 |
%\phi(f_\cJ(a_1, \cdots, a_k)) = f_{\phi(\cJ)}(\phi_1(a_1), \cdots, \phi_k(a_k)). |
|
1378 |
%\end{equation*} |
|
1379 |
%\end{enumerate} |
|
1380 |
%\end{defn} |
|
1381 |
||
345 | 1382 |
From a topological $A_\infty$ category on $[0,1]$ $\cC$ we can produce a `conventional' |
1383 |
$A_\infty$ category $(A, \{m_k\})$ as defined in, for example, \cite{MR1854636}. |
|
1384 |
We'll just describe the algebra case (that is, a category with only one object), |
|
1385 |
as the modifications required to deal with multiple objects are trivial. |
|
1386 |
Define $A = \cC$ as a chain complex (so $m_1 = d$). |
|
1387 |
Define $m_2 : A\tensor A \to A$ by $f_{\{(0,\frac{1}{2}),(\frac{1}{2},1)\}}$. |
|
1388 |
To define $m_3$, we begin by taking the one parameter family $\phi_3$ of diffeomorphisms |
|
1389 |
of $[0,1]$ that interpolates linearly between the identity and the piecewise linear |
|
1390 |
diffeomorphism taking $\frac{1}{4}$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $\frac{3}{4}$, and then define |
|
194 | 1391 |
\begin{equation*} |
1392 |
m_3(a,b,c) = ev(\phi_3, m_2(m_2(a,b), c)). |
|
1393 |
\end{equation*} |
|
1394 |
||
1395 |
It's then easy to calculate that |
|
1396 |
\begin{align*} |
|
1397 |
d(m_3(a,b,c)) & = ev(d \phi_3, m_2(m_2(a,b),c)) - ev(\phi_3 d m_2(m_2(a,b), c)) \\ |
|
1398 |
& = ev( \phi_3(1), m_2(m_2(a,b),c)) - ev(\phi_3(0), m_2 (m_2(a,b),c)) - \\ & \qquad - ev(\phi_3, m_2(m_2(da, b), c) + (-1)^{\deg a} m_2(m_2(a, db), c) + \\ & \qquad \quad + (-1)^{\deg a+\deg b} m_2(m_2(a, b), dc) \\ |
|
1399 |
& = m_2(a , m_2(b,c)) - m_2(m_2(a,b),c) - \\ & \qquad - m_3(da,b,c) + (-1)^{\deg a + 1} m_3(a,db,c) + \\ & \qquad \quad + (-1)^{\deg a + \deg b + 1} m_3(a,b,dc), \\ |
|
1400 |
\intertext{and thus that} |
|
1401 |
m_1 \circ m_3 & = m_2 \circ (\id \tensor m_2) - m_2 \circ (m_2 \tensor \id) - \\ & \qquad - m_3 \circ (m_1 \tensor \id \tensor \id) - m_3 \circ (\id \tensor m_1 \tensor \id) - m_3 \circ (\id \tensor \id \tensor m_1) |
|
1402 |
\end{align*} |
|
1403 |
as required (c.f. \cite[p. 6]{MR1854636}). |
|
1404 |
\todo{then the general case.} |
|
345 | 1405 |
We won't describe a reverse construction (producing a topological $A_\infty$ category |
477
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1406 |
from a ``conventional" $A_\infty$ category), but we presume that this will be easy for the experts. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1407 |
|
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
1408 |
} %%%%% end \noop %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |