author | Scott Morrison <scott@tqft.net> |
Thu, 18 Nov 2010 10:45:52 -0800 | |
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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 topological $n$-category from a traditional $n$-category; the morphisms of the |
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topological $n$-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 topological $n$-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 topological $n$-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 topological $n$-categories |
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(the paper is already too long!), 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) plain $n$-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 topological $1$-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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If the underlying manifolds for $\cX$ have further geometric structure, then we obtain certain functors. |
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The base case is for oriented manifolds, where we obtain no extra algebraic data. |
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For 1-categories based on unoriented manifolds, |
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there is a map $*:c(\cX)^1\to c(\cX)^1$ |
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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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Topological properties of this homeomorphism imply that |
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$a^{**} = a$ (* is order 2), * reverses domain and range, and $(ab)^* = b^*a^*$ |
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(* is an anti-automorphism). |
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For 1-categories based on Spin manifolds, |
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the 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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For 1-categories based on $\text{Pin}_-$ manifolds, |
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we have an order 4 antiautomorphism of $c(\cX)^1$. |
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For 1-categories based on $\text{Pin}_+$ manifolds, |
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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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%\nn{need to also consider automorphisms of $B^0$ / 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 topological |
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$1$-category $t(C)$. |
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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 topological 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 topological 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 topological 2-category. |
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We will construct from $\cC$ a traditional pivotal 2-category. |
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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 the 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 $C^k$ of $C$ to be $\cC(B^k)_E$, 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 $\cC(B^k)_E$ 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 $C$ 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 $C^1$. |
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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 $C^2$ (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 $a\in C^1$, define $\id_a = a\times I \in C^2$ (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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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 $\cC$. |
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This means we are free to add or delete product regions from 2-morphisms. |
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Let $a: y\to x$ be a 1-morphism. |
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Define 2-morphsims $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ |
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as shown in Figure \ref{fzo2}. |
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\begin{figure}[t] |
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\centering |
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\begin{tikzpicture} |
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\newcommand{\rr}{6} |
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\newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}} |
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\newcommand{\namedvertex}[1]{node[circle,fill=black,inner sep=1pt] (#1) {}} |
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\node(A) at (0,0) { |
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\begin{tikzpicture} |
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\node[red,left] at (0,0) {$y$}; |
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\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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\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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\begin{scope} |
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\path[clip] (0,0) arc (-120:-60:\rr) arc (60:120:\rr); |
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\foreach \x in {0,0.24,...,3} { |
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\draw[green!50!brown] (\x,1) -- (\x,-1); |
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} |
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\end{scope} |
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\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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\end{tikzpicture} |
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}; |
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\node(B) at (-4,-4) { |
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\begin{tikzpicture} |
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\node[red,left] at (0,0) {$y$}; |
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\draw (0,0) \vertex |
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arc (120:105:\rr) node[red,above] {$a$} |
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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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% 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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% arc (-90:-120:\rr); |
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\draw (0,0) |
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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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240 |
\draw (x1) -- node[red, above=3pt] {$x \times I$} (x2); |
498
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\begin{scope} |
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\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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245 |
} |
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246 |
\end{scope} |
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247 |
\end{tikzpicture} |
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248 |
}; |
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249 |
|
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250 |
\node(C) at (4,-4) { |
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251 |
\begin{tikzpicture}[y=-1cm] |
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252 |
\node[red,left] at (0,0) {$y$}; |
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|
253 |
\draw (0,0) \vertex |
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arc (120:105:\rr) node[red,below] {$a$} |
503
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|
255 |
arc (105:90:\rr) node[red,below] {$x$} \namedvertex{x1}; |
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256 |
% arc (90:75:\rr) node[red,below] {$x \times I$} |
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257 |
% arc (75:60:\rr) \vertex node[red,right] {$x$} |
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258 |
% arc (-60:-90:\rr) node[red,above] {$a$} |
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259 |
% arc (-90:-120:\rr); |
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260 |
\draw (0,0) |
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|
261 |
arc (-120:-90:\rr) node[red,above] {$a$} |
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|
262 |
arc (-90:-61:\rr) \namedvertex{x2} node[red,right] {$x$}; |
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|
263 |
\draw (x1) -- node[red, below=3pt] {$x \times I$} (x2); |
498
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264 |
\begin{scope} |
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265 |
\path[clip] (0,0) arc (-120:-60:\rr) arc (60:120:\rr); |
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266 |
\foreach \x in {0,0.48,...,9} { |
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267 |
\draw[green!50!brown] (\x/4,1) -- (\x,-1); |
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|
268 |
} |
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269 |
\end{scope} |
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270 |
\end{tikzpicture} |
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271 |
}; |
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272 |
|
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273 |
\draw[->] (A) -- (B); |
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|
274 |
\draw[->] (A) -- (C); |
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275 |
\end{tikzpicture} |
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|
276 |
\caption{Producing weak identities from half pinched products} |
126 | 277 |
\label{fzo2} |
278 |
\end{figure} |
|
457
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|
279 |
As suggested by the figure, these are two different reparameterizations |
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|
280 |
of a half-pinched version of $a\times I$. |
125 | 281 |
We must show that the two compositions of these two maps give the identity 2-morphisms |
282 |
on $a$ and $a\bullet \id_x$, as defined above. |
|
283 |
Figure \ref{fzo3} shows one case. |
|
126 | 284 |
\begin{figure}[t] |
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285 |
\centering |
508 | 286 |
\begin{tikzpicture} |
287 |
||
288 |
\newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}} |
|
289 |
\newcommand{\nsep}{1.8} |
|
290 |
||
291 |
\node(A) at (0,0) { |
|
292 |
\begin{tikzpicture} |
|
293 |
||
294 |
\draw (0,0) coordinate (p1); |
|
295 |
\draw (3.6,0) coordinate (p2); |
|
296 |
\draw (2.3,1) coordinate (p3); |
|
297 |
\draw (2.3,-1) coordinate (p4); |
|
298 |
||
299 |
\begin{scope} |
|
300 |
\clip (p1) .. controls +(.5,-.5) and +(-.8,0) .. (p4) -- |
|
301 |
(p2) -- (p3) .. controls +(-.8,0) and +(.5,.5) .. (p1); |
|
302 |
\foreach \x in {0,0.26,...,4} { |
|
303 |
\draw[green!50!brown] (\x,0) -- (intersection cs: first line={(p2)--(p3)}, second line={(0,0)--(0,1)}); |
|
304 |
\draw[green!50!brown] (\x,0) -- (intersection cs: first line={(p2)--(p4)}, second line={(0,0)--(0,1)}); |
|
305 |
} |
|
306 |
\end{scope} |
|
307 |
||
308 |
\draw (p1) .. controls ($(p1) + (.5,.5)$) and ($(p3) + (-.8,0)$) .. node[red, above=3pt] {$a$} (p3); |
|
309 |
\draw (p1) .. controls ($(p1) + (.5,-.5)$) and ($(p4) + (-.8,0)$) .. node[red, below=3pt] {$a$} (p4); |
|
310 |
\draw (p3) -- node[red, above=7pt, right=1pt] {$x \times I$} (p2); |
|
311 |
\draw (p4) -- node[red, below=7pt, right=1pt] {$x \times I$} (p2); |
|
312 |
\draw (p1) -- (p2); |
|
313 |
||
314 |
\draw (p1) \vertex; |
|
315 |
\draw (p2) \vertex; |
|
316 |
\draw (p3) \vertex; |
|
317 |
\draw (p4) \vertex; |
|
318 |
||
319 |
\end{tikzpicture} |
|
320 |
}; |
|
321 |
||
322 |
\node[outer sep=\nsep](B) at (5.5,0) { |
|
323 |
\begin{tikzpicture} |
|
324 |
||
325 |
\draw (0,0) coordinate (p1); |
|
326 |
\draw (3.6,0) coordinate (p2); |
|
327 |
\draw (2.3,1) coordinate (p3); |
|
328 |
\draw (2.3,-1) coordinate (p4); |
|
329 |
\draw (4.6,0) coordinate (p2b); |
|
330 |
||
331 |
\begin{scope} |
|
332 |
\clip (p1) .. controls +(.5,-.5) and +(-.8,0) .. (p4) -- |
|
333 |
(p2) -- (p3) .. controls +(-.8,0) and +(.5,.5) .. (p1); |
|
334 |
\foreach \x in {0,0.26,...,4} { |
|
335 |
\draw[green!50!brown] (\x,0) -- (intersection cs: first line={(p2)--(p3)}, second line={(0,0)--(0,1)}); |
|
336 |
\draw[green!50!brown] (\x,0) -- (intersection cs: first line={(p2)--(p4)}, second line={(0,0)--(0,1)}); |
|
337 |
} |
|
338 |
\end{scope} |
|
339 |
||
340 |
\begin{scope} |
|
341 |
\clip (p3)--(p2)--(p4)--(p2b)--cycle; |
|
342 |
\draw[blue!50!brown, step=.23] ($(p4)+(0,-1)$) grid +(3,3); |
|
343 |
\end{scope} |
|
344 |
||
345 |
\draw (p1) .. controls ($(p1) + (.5,.5)$) and ($(p3) + (-.8,0)$) .. node[red, above=3pt] {$a$} (p3); |
|
346 |
\draw (p1) .. controls ($(p1) + (.5,-.5)$) and ($(p4) + (-.8,0)$) .. node[red, below=3pt] {$a$} (p4); |
|
347 |
\draw (p3) -- (p2); |
|
348 |
\draw (p4) -- (p2); |
|
349 |
\draw (p3) -- node[red, above=7pt, right=1pt] {$x \times I$} (p2b); |
|
350 |
\draw (p4) -- node[red, below=7pt, right=1pt] {$x \times I$} (p2b); |
|
351 |
||
352 |
\draw (p1) \vertex; |
|
353 |
\draw (p2) \vertex; |
|
354 |
\draw (p3) \vertex; |
|
355 |
\draw (p4) \vertex; |
|
356 |
\draw (p2b) \vertex; |
|
357 |
||
358 |
\end{tikzpicture} |
|
359 |
}; |
|
360 |
||
361 |
\node[outer sep=\nsep](C) at (11,0) { |
|
362 |
\begin{tikzpicture} |
|
363 |
||
364 |
\draw (0,0) coordinate (p1); |
|
365 |
\draw (2.3,0) coordinate (p2); |
|
366 |
\draw (2.3,1) coordinate (p3); |
|
367 |
\draw (2.3,-1) coordinate (p4); |
|
368 |
\draw (3.6,0) coordinate (p2b); |
|
369 |
||
370 |
\begin{scope} |
|
371 |
\clip (p1) .. controls +(.5,-.5) and +(-.8,0) .. (p4) -- |
|
372 |
(p2) -- (p3) .. controls +(-.8,0) and +(.5,.5) .. (p1); |
|
373 |
\foreach \x in {0,0.26,...,4} { |
|
374 |
\draw[green!50!brown] (\x,-1) -- (\x,1); |
|
375 |
} |
|
376 |
\end{scope} |
|
377 |
||
378 |
\begin{scope} |
|
379 |
\clip (p3)--(p2)--(p4)--(p2b)--cycle; |
|
380 |
\draw[blue!50!brown, step=.23] ($(p4)+(0,-1)$) grid +(3,3); |
|
381 |
\end{scope} |
|
382 |
||
383 |
\draw (p1) .. controls ($(p1) + (.5,.5)$) and ($(p3) + (-.8,0)$) .. node[red, above=3pt] {$a$} (p3); |
|
384 |
\draw (p1) .. controls ($(p1) + (.5,-.5)$) and ($(p4) + (-.8,0)$) .. node[red, below=3pt] {$a$} (p4); |
|
385 |
\draw[green!50!brown] (p3) -- (p4); |
|
386 |
\draw (p3) -- node[red, above=7pt, right=1pt] {$x \times I$} (p2b); |
|
387 |
\draw (p4) -- node[red, below=7pt, right=1pt] {$x \times I$} (p2b); |
|
388 |
||
389 |
\draw (p1) \vertex; |
|
390 |
\draw (p3) \vertex; |
|
391 |
\draw (p4) \vertex; |
|
392 |
\draw (p2b) \vertex; |
|
393 |
||
394 |
\end{tikzpicture} |
|
395 |
}; |
|
396 |
||
397 |
\draw[->, thick, blue!50!green] (A) -- (B); |
|
398 |
\draw[->, thick, blue!50!green] (B) -- node[black, above] {$=$} (C); |
|
399 |
||
400 |
\end{tikzpicture} |
|
498
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|
401 |
\caption{Composition of weak identities, 1} |
126 | 402 |
\label{fzo3} |
403 |
\end{figure} |
|
457
54328be726e7
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diff
changeset
|
404 |
In the first step we have inserted a copy of $(x\times I)\times I$. |
125 | 405 |
Figure \ref{fzo4} shows the other case. |
126 | 406 |
\begin{figure}[t] |
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changeset
|
407 |
\centering |
509 | 408 |
\begin{tikzpicture} |
409 |
||
410 |
\newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}} |
|
411 |
\newcommand{\nsep}{1.8} |
|
412 |
||
510 | 413 |
\clip (-4,-1.25)--(12,-1.25)--(16,1.25)--(-1,1.25)--cycle; |
414 |
||
415 |
||
509 | 416 |
\node[outer sep=\nsep](A) at (0,0) { |
417 |
\begin{tikzpicture} |
|
418 |
\draw (0,0) coordinate (p1); |
|
419 |
\draw (4,0) coordinate (p2); |
|
420 |
\draw (2.4,0) coordinate (p2a); |
|
421 |
\draw (2,1.2) coordinate (pu); |
|
422 |
\draw (2,-1.2) coordinate (pd); |
|
423 |
||
424 |
\begin{scope} |
|
425 |
\clip (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
426 |
\foreach \t in {0,.065,...,1} { |
|
427 |
\draw[green!50!brown] ($(p1)!\t!(p2a)$) -- +(90 - \t*90 + \t*6 : 4); |
|
428 |
\draw[green!50!brown] ($(p1)!\t!(p2a)$) -- +(-90 + \t*90 - \t*6 : 4); |
|
429 |
} |
|
430 |
\draw[dashed] ($(p2a) + (-.6,3)$)--(p2a)--($(p2a) + (-.6,-3)$); |
|
431 |
\end{scope} |
|
432 |
||
433 |
\draw (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
434 |
\draw (p1)--(p2); |
|
435 |
||
436 |
\draw (p1) \vertex; |
|
437 |
\draw (p2) \vertex; |
|
438 |
\draw (p2a) \vertex; |
|
439 |
\end{tikzpicture} |
|
440 |
}; |
|
441 |
||
442 |
\node[outer sep=\nsep](B) at (5,0) { |
|
443 |
\begin{tikzpicture} |
|
444 |
\draw (0,0) coordinate (p1); |
|
445 |
\draw (4,0) coordinate (p2); |
|
446 |
\draw (2.4,0) coordinate (p2a); |
|
447 |
\draw (2,1.2) coordinate (pu); |
|
448 |
\draw (2,-1.2) coordinate (pd); |
|
449 |
||
450 |
\begin{scope} |
|
451 |
\clip (-.1,3)--($(p2a) + (-.6,3)$)--(p2a)--($(p2a) + (-.6,-3)$)--(-.1,-3)--cycle; |
|
452 |
||
453 |
\begin{scope} |
|
454 |
\clip (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
455 |
\foreach \t in {0,.065,...,1} { |
|
456 |
\draw[green!50!brown] ($(p1)!\t!(p2a)$) -- +(90 - \t*90 + \t*6 : 4); |
|
457 |
\draw[green!50!brown] ($(p1)!\t!(p2a)$) -- +(-90 + \t*90 - \t*6 : 4); |
|
458 |
} |
|
459 |
\draw ($(p2a) + (-.6,3)$)--(p2a)--($(p2a) + (-.6,-3)$); |
|
460 |
\end{scope} |
|
461 |
||
462 |
||
463 |
\draw (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
464 |
%\draw (p1)--(p2); |
|
465 |
\end{scope} |
|
466 |
||
467 |
\begin{scope} |
|
468 |
\clip (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
469 |
\draw ($(p2a) + (-.6,3)$)--(p2a)--($(p2a) + (-.6,-3)$); |
|
470 |
\end{scope} |
|
471 |
||
472 |
\draw (p1) \vertex; |
|
473 |
\draw (p2a) \vertex; |
|
474 |
\end{tikzpicture} |
|
475 |
}; |
|
476 |
||
477 |
\node[outer sep=\nsep](C) at (9,0) { |
|
478 |
\begin{tikzpicture} |
|
479 |
\draw (0,0) coordinate (p1); |
|
480 |
\draw (4,0) coordinate (p2); |
|
481 |
\draw (2,1.2) coordinate (pu); |
|
482 |
\draw (2,-1.2) coordinate (pd); |
|
483 |
||
484 |
\begin{scope} |
|
485 |
\clip (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
486 |
\foreach \t in {0,.045,...,1} { |
|
487 |
\draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); |
|
488 |
} |
|
489 |
\end{scope} |
|
490 |
||
491 |
\draw (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
492 |
||
493 |
\draw (p1) \vertex; |
|
494 |
\draw (p2) \vertex; |
|
495 |
\end{tikzpicture} |
|
496 |
}; |
|
497 |
||
498 |
\draw[->, thick, blue!50!green] (A) -- (B); |
|
499 |
\draw[->, thick, blue!50!green] ($(B) + (1,0)$) -- node[black, above] {$=$} (C); |
|
500 |
||
501 |
\end{tikzpicture} |
|
498
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|
502 |
\caption{Composition of weak identities, 2} |
126 | 503 |
\label{fzo4} |
504 |
\end{figure} |
|
457
54328be726e7
comparing_defs.tex 2-cat section
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451
diff
changeset
|
505 |
We identify a product region and remove it. |
124 | 506 |
|
510 | 507 |
We define horizontal composition $f *_h g$ of 2-morphisms $f$ and $g$ as shown in Figure \ref{fzo5}. |
345 | 508 |
It is not hard to show that this is independent of the arbitrary (left/right) |
509 |
choice made in the definition, and that it is associative. |
|
127 | 510 |
\begin{figure}[t] |
511 |
\begin{equation*} |
|
510 | 512 |
\raisebox{-.9cm}{ |
513 |
\begin{tikzpicture} |
|
514 |
\draw (0,0) .. controls +(1,.8) and +(-1,.8) .. node[above] {$b$} (2.9,0) |
|
515 |
.. controls +(-1,-.8) and +(1,-.8) .. node[below] {$a$} (0,0); |
|
516 |
\draw[->, thick, orange!50!brown] (1.45,-.4)-- node[left, black] {$f$} +(0,.8); |
|
517 |
\end{tikzpicture}} |
|
518 |
\;\;\;*_h\;\; |
|
519 |
\raisebox{-.9cm}{ |
|
520 |
\begin{tikzpicture} |
|
521 |
\draw (0,0) .. controls +(1,.8) and +(-1,.8) .. node[above] {$d$} (2.9,0) |
|
522 |
.. controls +(-1,-.8) and +(1,-.8) .. node[below] {$c$} (0,0); |
|
523 |
\draw[->, thick, orange!50!brown] (1.45,-.4)-- node[left, black] {$g$} +(0,.8); |
|
524 |
\end{tikzpicture}} |
|
525 |
\;=\; |
|
526 |
\raisebox{-1.9cm}{ |
|
527 |
\begin{tikzpicture} |
|
528 |
\draw (0,0) coordinate (p1); |
|
529 |
\draw (5.8,0) coordinate (p2); |
|
530 |
\draw (2.9,.3) coordinate (pu); |
|
531 |
\draw (2.9,-.3) coordinate (pd); |
|
532 |
\begin{scope} |
|
533 |
\clip (p1) .. controls +(.6,.3) and +(-.5,0) .. (pu) |
|
534 |
.. controls +(.5,0) and +(-.6,.3) .. (p2) |
|
535 |
.. controls +(-.6,-.3) and +(.5,0) .. (pd) |
|
536 |
.. controls +(-.5,0) and +(.6,-.3) .. (p1); |
|
537 |
\foreach \t in {0,.03,...,1} { |
|
538 |
\draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); |
|
539 |
} |
|
540 |
\end{scope} |
|
541 |
\draw (p1) .. controls +(.6,.3) and +(-.5,0) .. (pu) |
|
542 |
.. controls +(.5,0) and +(-.6,.3) .. (p2) |
|
543 |
.. controls +(-.6,-.3) and +(.5,0) .. (pd) |
|
544 |
.. controls +(-.5,0) and +(.6,-.3) .. (p1); |
|
545 |
\draw (p1) .. controls +(1,-2) and +(-1,-1) .. (pd); |
|
546 |
\draw (p2) .. controls +(-1,2) and +(1,1) .. (pu); |
|
547 |
\draw[->, thick, orange!50!brown] (1.45,-1.1)-- node[left, black] {$f$} +(0,.7); |
|
548 |
\draw[->, thick, orange!50!brown] (4.35,.4)-- node[left, black] {$g$} +(0,.7); |
|
549 |
\draw[->, thick, blue!75!yellow] (1.5,.78) node[black, above] {$(b\cdot c)\times I$} -- (2.5,0); |
|
550 |
\end{tikzpicture}} |
|
127 | 551 |
\end{equation*} |
552 |
\caption{Horizontal composition of 2-morphisms} |
|
553 |
\label{fzo5} |
|
554 |
\end{figure} |
|
125 | 555 |
|
457
54328be726e7
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|
556 |
%\nn{need to find a list of axioms for pivotal 2-cats to check} |
114 | 557 |
|
194 | 558 |
|
512
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fixing some (but not all!?) of the hyperref warnings; start on revision of evmap
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|
559 |
\subsection{\texorpdfstring{$A_\infty$}{A-infinity} 1-categories} |
194 | 560 |
\label{sec:comparing-A-infty} |
431 | 561 |
In this section, we make contact between the usual definition of an $A_\infty$ category |
477
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|
562 |
and our definition of a topological $A_\infty$ $1$-category, from \S \ref{ss:n-cat-def}. |
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|
563 |
|
86c8e2129355
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|
564 |
\medskip |
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|
565 |
|
480 | 566 |
Given a topological $A_\infty$ $1$-category $\cC$, we define an ``$m_k$-style" |
477
86c8e2129355
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changeset
|
567 |
$A_\infty$ $1$-category $A$ as follows. |
86c8e2129355
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changeset
|
568 |
The objects of $A$ are $\cC(pt)$. |
86c8e2129355
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diff
changeset
|
569 |
The morphisms of $A$, from $x$ to $y$, are $\cC(I; x, y)$ |
86c8e2129355
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changeset
|
570 |
($\cC$ applied to the standard interval with boundary labeled by $x$ and $y$). |
86c8e2129355
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|
571 |
For simplicity we will now assume there is only one object and suppress it from the notation. |
86c8e2129355
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diff
changeset
|
572 |
|
86c8e2129355
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changeset
|
573 |
A choice of homeomorphism $I\cup I \to I$ induces a chain map $m_2: A\times A\to A$. |
86c8e2129355
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|
574 |
We now have two different homeomorphisms $I\cup I\cup I \to I$, but they are isotopic. |
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|
575 |
Choose a specific 1-parameter family of homeomorphisms connecting them; this induces |
86c8e2129355
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changeset
|
576 |
a degree 1 chain homotopy $m_3:A\ot A\ot A\to A$. |
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changeset
|
577 |
Proceeding in this way we define the rest of the $m_i$'s. |
86c8e2129355
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|
578 |
It is straightforward to verify that they satisfy the necessary identities. |
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|
579 |
|
86c8e2129355
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|
580 |
\medskip |
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|
581 |
|
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|
582 |
In the other direction, we start with an alternative conventional definition of an $A_\infty$ algebra: |
86c8e2129355
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|
583 |
an algebra $A$ for the $A_\infty$ operad. |
86c8e2129355
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|
584 |
(For simplicity, we are assuming our $A_\infty$ 1-category has only one object.) |
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|
585 |
We are free to choose any operad with contractible spaces, so we choose the operad |
86c8e2129355
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|
586 |
whose $k$-th space is the space of decompositions of the standard interval $I$ into $k$ |
86c8e2129355
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diff
changeset
|
587 |
parameterized copies of $I$. |
480 | 588 |
Note in particular that when $k=1$ this implies a $C_*(\Homeo(I))$ action on $A$. |
589 |
(Compare with Example \ref{ex:e-n-alg} and the discussion which precedes it.) |
|
477
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|
590 |
Given a non-standard interval $J$, we define $\cC(J)$ to be |
86c8e2129355
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diff
changeset
|
591 |
$(\Homeo(I\to J) \times A)/\Homeo(I\to I)$, |
86c8e2129355
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diff
changeset
|
592 |
where $\beta \in \Homeo(I\to I)$ acts via $(f, a) \mapsto (f\circ \beta\inv, \beta_*(a))$. |
480 | 593 |
Note that $\cC(J) \cong A$ (non-canonically) for all intervals $J$. |
477
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diff
changeset
|
594 |
We define a $\Homeo(J)$ action on $\cC(J)$ via $g_*(f, a) = (g\circ f, a)$. |
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changeset
|
595 |
The $C_*(\Homeo(J))$ action is defined similarly. |
86c8e2129355
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diff
changeset
|
596 |
|
86c8e2129355
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|
597 |
Let $J_1$ and $J_2$ be intervals. |
86c8e2129355
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diff
changeset
|
598 |
We must define a map $\cC(J_1)\ot\cC(J_2)\to\cC(J_1\cup J_2)$. |
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diff
changeset
|
599 |
Choose a homeomorphism $g:I\to J_1\cup J_2$. |
86c8e2129355
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diff
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|
600 |
Let $(f_i, a_i)\in \cC(J_i)$. |
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diff
changeset
|
601 |
We have a parameterized decomposition of $I$ into two intervals given by |
86c8e2129355
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diff
changeset
|
602 |
$g\inv \circ f_i$, $i=1,2$. |
86c8e2129355
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diff
changeset
|
603 |
Corresponding to this decomposition the operad action gives a map $\mu: A\ot A\to A$. |
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diff
changeset
|
604 |
Define the gluing map to send $(f_1, a_1)\ot (f_2, a_2)$ to $(g, \mu(a_1\ot a_2))$. |
480 | 605 |
Operad associativity for $A$ implies that this gluing map is independent of the choice of |
606 |
$g$ and the choice of representative $(f_i, a_i)$. |
|
477
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changeset
|
607 |
|
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changeset
|
608 |
It is straightforward to verify the remaining axioms for a topological $A_\infty$ 1-category. |
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|
609 |
|
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changeset
|
610 |
|
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changeset
|
611 |
|
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|
612 |
|
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|
613 |
|
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|
614 |
|
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|
615 |
|
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|
616 |
\noop { %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
194 | 617 |
|
431 | 618 |
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 | 619 |
\begin{defn} |
345 | 620 |
A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$, |
621 |
and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with |
|
194 | 622 |
\begin{itemize} |
623 |
\item an action of the operad of $\Obj(\cC)$-labeled cell decompositions |
|
624 |
\item and a compatible action of $\CD{[0,1]}$. |
|
625 |
\end{itemize} |
|
626 |
\end{defn} |
|
345 | 627 |
Here the operad of cell decompositions of $[0,1]$ has operations indexed by a finite set of |
628 |
points $0 < x_1< \cdots < x_k < 1$, cutting $[0,1]$ into subintervals. |
|
629 |
An $X$-labeled cell decomposition labels $\{0, x_1, \ldots, x_k, 1\}$ by $X$. |
|
630 |
Given two cell decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$, we can compose |
|
631 |
them to form a new cell decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$ by inserting the points |
|
632 |
of $\cJ^{(2)}$ linearly into the $m$-th interval of $\cJ^{(1)}$. |
|
633 |
In the $X$-labeled case, we insist that the appropriate labels match up. |
|
634 |
Saying we have an action of this operad means that for each labeled cell decomposition |
|
635 |
$0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain |
|
431 | 636 |
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 | 637 |
chain maps compose exactly as the cell decompositions. |
638 |
An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad |
|
639 |
if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which |
|
640 |
is supported on the subintervals determined by $\pi$, then the two possible operations |
|
641 |
(glue intervals together, then apply the diffeomorphisms, or apply the diffeormorphisms |
|
642 |
separately to the subintervals, then glue) commute (as usual, up to a weakly unique homotopy). |
|
194 | 643 |
|
345 | 644 |
Translating between this notion and the usual definition of an $A_\infty$ category is now straightforward. |
645 |
To restrict to the standard interval, define $\cC_{a,b} = \cC([0,1];a,b)$. |
|
646 |
Given a cell decomposition $0 < x_1< \cdots < x_k < 1$, we use the map (suppressing labels) |
|
194 | 647 |
$$\cC([0,1])^{\tensor k+1} \to \cC([0,x_1]) \tensor \cdots \tensor \cC[x_k,1] \to \cC([0,1])$$ |
345 | 648 |
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. |
649 |
The action of $\CD{[0,1]}$ carries across, and is automatically compatible. |
|
650 |
Going the other way, we just declare $\cC(J;a,b) = \cC_{a,b}$, pick a diffeomorphism |
|
651 |
$\phi_J : J \isoto [0,1]$ for every interval $J$, define the gluing map |
|
652 |
$\cC(J_1) \tensor \cC(J_2) \to \cC(J_1 \cup J_2)$ by the first applying |
|
653 |
the cell decomposition map for $0 < \frac{1}{2} < 1$, then the self-diffeomorphism of $[0,1]$ |
|
654 |
given by $\frac{1}{2} (\phi_{J_1} \cup (1+ \phi_{J_2})) \circ \phi_{J_1 \cup J_2}^{-1}$. |
|
655 |
You can readily check that this gluing map is associative on the nose. \todo{really?} |
|
194 | 656 |
|
657 |
%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 $\cJ^{(1)} \circ_m \cJ^{(2)}$. |
|
658 |
||
659 |
%\begin{defn} |
|
660 |
%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'. |
|
661 |
||
662 |
%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 |
|
663 |
%\begin{equation*} |
|
664 |
%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}). |
|
665 |
%\end{equation*} |
|
666 |
||
667 |
%An \emph{action of families of diffeomorphisms} is a chain map $ev: \CD{[0,1]} \tensor A \to A$, such that |
|
668 |
%\begin{enumerate} |
|
669 |
%\item The diagram |
|
670 |
%\begin{equation*} |
|
671 |
%\xymatrix{ |
|
672 |
%\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} \\ |
|
673 |
%\CD{[0,1]} \tensor A \ar[r]^{ev} & A |
|
674 |
%} |
|
675 |
%\end{equation*} |
|
676 |
%commutes up to weakly unique homotopy. |
|
677 |
%\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 |
|
678 |
%\begin{equation*} |
|
679 |
%\phi(f_\cJ(a_1, \cdots, a_k)) = f_{\phi(\cJ)}(\phi_1(a_1), \cdots, \phi_k(a_k)). |
|
680 |
%\end{equation*} |
|
681 |
%\end{enumerate} |
|
682 |
%\end{defn} |
|
683 |
||
345 | 684 |
From a topological $A_\infty$ category on $[0,1]$ $\cC$ we can produce a `conventional' |
685 |
$A_\infty$ category $(A, \{m_k\})$ as defined in, for example, \cite{MR1854636}. |
|
686 |
We'll just describe the algebra case (that is, a category with only one object), |
|
687 |
as the modifications required to deal with multiple objects are trivial. |
|
688 |
Define $A = \cC$ as a chain complex (so $m_1 = d$). |
|
689 |
Define $m_2 : A\tensor A \to A$ by $f_{\{(0,\frac{1}{2}),(\frac{1}{2},1)\}}$. |
|
690 |
To define $m_3$, we begin by taking the one parameter family $\phi_3$ of diffeomorphisms |
|
691 |
of $[0,1]$ that interpolates linearly between the identity and the piecewise linear |
|
692 |
diffeomorphism taking $\frac{1}{4}$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $\frac{3}{4}$, and then define |
|
194 | 693 |
\begin{equation*} |
694 |
m_3(a,b,c) = ev(\phi_3, m_2(m_2(a,b), c)). |
|
695 |
\end{equation*} |
|
696 |
||
697 |
It's then easy to calculate that |
|
698 |
\begin{align*} |
|
699 |
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)) \\ |
|
700 |
& = 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) \\ |
|
701 |
& = 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), \\ |
|
702 |
\intertext{and thus that} |
|
703 |
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) |
|
704 |
\end{align*} |
|
705 |
as required (c.f. \cite[p. 6]{MR1854636}). |
|
706 |
\todo{then the general case.} |
|
345 | 707 |
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
|
708 |
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
|
709 |
|
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
710 |
} %%%%% end \noop %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |