author | Scott Morrison <scott@tqft.net> |
Fri, 23 Jul 2010 13:52:30 -0700 | |
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parent 477 | 86c8e2129355 |
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%!TEX root = ../../blob1.tex |
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\section{Comparing $n$-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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\nn{say something about modules and tensor products?} |
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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 $\Set$ 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{Plain 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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\begin{equation*} |
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\mathfig{.73}{tempkw/zo1} |
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\end{equation*} |
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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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\begin{equation*} |
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\mathfig{.73}{tempkw/zo2} |
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\end{equation*} |
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\caption{blah blah} |
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\label{fzo2} |
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\end{figure} |
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As suggested by the figure, these are two different reparameterizations |
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of a half-pinched version of $a\times I$. |
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We must show that the two compositions of these two maps give the identity 2-morphisms |
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on $a$ and $a\bullet \id_x$, as defined above. |
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Figure \ref{fzo3} shows one case. |
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\begin{figure}[t] |
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\begin{equation*} |
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\mathfig{.83}{tempkw/zo3} |
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\end{equation*} |
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\caption{blah blah} |
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\label{fzo3} |
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\end{figure} |
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In the first step we have inserted a copy of $(x\times I)\times I$. |
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Figure \ref{fzo4} shows the other case. |
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\begin{figure}[t] |
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\begin{equation*} |
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\mathfig{.83}{tempkw/zo4} |
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\end{equation*} |
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\caption{blah blah} |
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\label{fzo4} |
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\end{figure} |
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We identify a product region and remove it. |
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We define horizontal composition of 2-morphisms as shown in Figure \ref{fzo5}. |
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It is not hard to show that this is independent of the arbitrary (left/right) |
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choice made in the definition, and that it is associative. |
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\begin{figure}[t] |
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\begin{equation*} |
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\mathfig{.83}{tempkw/zo5} |
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\end{equation*} |
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\caption{Horizontal composition of 2-morphisms} |
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\label{fzo5} |
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\end{figure} |
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%\nn{need to find a list of axioms for pivotal 2-cats to check} |
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\subsection{$A_\infty$ $1$-categories} |
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\label{sec:comparing-A-infty} |
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In this section, we make contact between the usual definition of an $A_\infty$ category |
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and our definition of a topological $A_\infty$ $1$-category, from \S \ref{ss:n-cat-def}. |
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\medskip |
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Given a topological $A_\infty$ $1$-category $\cC$, we define an ``$m_k$-style |
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$A_\infty$ $1$-category $A$ as follows. |
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The objects of $A$ are $\cC(pt)$. |
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The morphisms of $A$, from $x$ to $y$, are $\cC(I; x, y)$ |
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($\cC$ applied to the standard interval with boundary labeled by $x$ and $y$). |
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For simplicity we will now assume there is only one object and suppress it from the notation. |
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A choice of homeomorphism $I\cup I \to I$ induces a chain map $m_2: A\times A\to A$. |
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We now have two different homeomorphisms $I\cup I\cup I \to I$, but they are isotopic. |
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Choose a specific 1-parameter family of homeomorphisms connecting them; this induces |
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a degree 1 chain homotopy $m_3:A\ot A\ot A\to A$. |
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Proceeding in this way we define the rest of the $m_i$'s. |
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It is straightforward to verify that they satisfy the necessary identities. |
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\medskip |
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In the other direction, we start with an alternative conventional definition of an $A_\infty$ algebra: |
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an algebra $A$ for the $A_\infty$ operad. |
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(For simplicity, we are assuming our $A_\infty$ 1-category has only one object.) |
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We are free to choose any operad with contractible spaces, so we choose the operad |
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whose $k$-th space is the space of decompositions of the standard interval $I$ into $k$ |
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parameterized copies of $I$. |
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Note in particular that when $k=1$ this implies a $\Homeo(I)$ action on $A$. |
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(Compare with Example \ref{ex:e-n-alg} and preceding discussion.) |
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Given a non-standard interval $J$, we define $\cC(J)$ to be |
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$(\Homeo(I\to J) \times A)/\Homeo(I\to I)$, |
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where $\beta \in \Homeo(I\to I)$ acts via $(f, a) \mapsto (f\circ \beta\inv, \beta_*(a))$. |
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\nn{check this} |
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We define a $\Homeo(J)$ action on $\cC(J)$ via $g_*(f, a) = (g\circ f, a)$. |
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The $C_*(\Homeo(J))$ action is defined similarly. |
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Let $J_1$ and $J_2$ be intervals. |
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We must define a map $\cC(J_1)\ot\cC(J_2)\to\cC(J_1\cup J_2)$. |
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Choose a homeomorphism $g:I\to J_1\cup J_2$. |
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Let $(f_i, a_i)\in \cC(J_i)$. |
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We have a parameterized decomposition of $I$ into two intervals given by |
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$g\inv \circ f_i$, $i=1,2$. |
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Corresponding to this decomposition the operad action gives a map $\mu: A\ot A\to A$. |
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Define the gluing map to send $(f_1, a_1)\ot (f_2, a_2)$ to $(g, \mu(a_1\ot a_2))$. |
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It is straightforward to verify the remaining axioms for a topological $A_\infty$ 1-category. |
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\noop { %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
194 | 263 |
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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 $[0,1]$. |
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\begin{defn} |
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A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$, |
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and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with |
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\begin{itemize} |
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\item an action of the operad of $\Obj(\cC)$-labeled cell decompositions |
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\item and a compatible action of $\CD{[0,1]}$. |
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\end{itemize} |
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\end{defn} |
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Here the operad of cell decompositions of $[0,1]$ has operations indexed by a finite set of |
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points $0 < x_1< \cdots < x_k < 1$, cutting $[0,1]$ into subintervals. |
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An $X$-labeled cell decomposition labels $\{0, x_1, \ldots, x_k, 1\}$ by $X$. |
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Given two cell decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$, we can compose |
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them to form a new cell decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$ by inserting the points |
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of $\cJ^{(2)}$ linearly into the $m$-th interval of $\cJ^{(1)}$. |
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In the $X$-labeled case, we insist that the appropriate labels match up. |
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Saying we have an action of this operad means that for each labeled cell decomposition |
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$0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain |
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map $$\cC_{a_0,a_1} \tensor \cdots \tensor \cC_{a_k,a_{k+1}} \to \cC_{a_0,a_{k+1}}$$ and these |
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chain maps compose exactly as the cell decompositions. |
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An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad |
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if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which |
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is supported on the subintervals determined by $\pi$, then the two possible operations |
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(glue intervals together, then apply the diffeomorphisms, or apply the diffeormorphisms |
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separately to the subintervals, then glue) commute (as usual, up to a weakly unique homotopy). |
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Translating between this notion and the usual definition of an $A_\infty$ category is now straightforward. |
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To restrict to the standard interval, define $\cC_{a,b} = \cC([0,1];a,b)$. |
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Given a cell decomposition $0 < x_1< \cdots < x_k < 1$, we use the map (suppressing labels) |
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$$\cC([0,1])^{\tensor k+1} \to \cC([0,x_1]) \tensor \cdots \tensor \cC[x_k,1] \to \cC([0,1])$$ |
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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. |
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The action of $\CD{[0,1]}$ carries across, and is automatically compatible. |
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Going the other way, we just declare $\cC(J;a,b) = \cC_{a,b}$, pick a diffeomorphism |
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$\phi_J : J \isoto [0,1]$ for every interval $J$, define the gluing map |
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$\cC(J_1) \tensor \cC(J_2) \to \cC(J_1 \cup J_2)$ by the first applying |
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the cell decomposition map for $0 < \frac{1}{2} < 1$, then the self-diffeomorphism of $[0,1]$ |
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given by $\frac{1}{2} (\phi_{J_1} \cup (1+ \phi_{J_2})) \circ \phi_{J_1 \cup J_2}^{-1}$. |
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You can readily check that this gluing map is associative on the nose. \todo{really?} |
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%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 $\cJ^{(1)} \circ_m \cJ^{(2)}$. |
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||
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%\begin{defn} |
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%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'. |
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||
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%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 |
|
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%\begin{equation*} |
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%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}). |
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%\end{equation*} |
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||
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%An \emph{action of families of diffeomorphisms} is a chain map $ev: \CD{[0,1]} \tensor A \to A$, such that |
|
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%\begin{enumerate} |
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%\item The diagram |
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%\begin{equation*} |
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%\xymatrix{ |
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%\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} \\ |
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%\CD{[0,1]} \tensor A \ar[r]^{ev} & A |
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%} |
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%\end{equation*} |
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%commutes up to weakly unique homotopy. |
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%\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 |
|
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%\begin{equation*} |
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%\phi(f_\cJ(a_1, \cdots, a_k)) = f_{\phi(\cJ)}(\phi_1(a_1), \cdots, \phi_k(a_k)). |
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%\end{equation*} |
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%\end{enumerate} |
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%\end{defn} |
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||
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From a topological $A_\infty$ category on $[0,1]$ $\cC$ we can produce a `conventional' |
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$A_\infty$ category $(A, \{m_k\})$ as defined in, for example, \cite{MR1854636}. |
|
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We'll just describe the algebra case (that is, a category with only one object), |
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as the modifications required to deal with multiple objects are trivial. |
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Define $A = \cC$ as a chain complex (so $m_1 = d$). |
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Define $m_2 : A\tensor A \to A$ by $f_{\{(0,\frac{1}{2}),(\frac{1}{2},1)\}}$. |
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To define $m_3$, we begin by taking the one parameter family $\phi_3$ of diffeomorphisms |
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of $[0,1]$ that interpolates linearly between the identity and the piecewise linear |
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diffeomorphism taking $\frac{1}{4}$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $\frac{3}{4}$, and then define |
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\begin{equation*} |
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m_3(a,b,c) = ev(\phi_3, m_2(m_2(a,b), c)). |
|
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\end{equation*} |
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||
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It's then easy to calculate that |
|
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\begin{align*} |
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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)) \\ |
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& = 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) \\ |
|
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& = 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), \\ |
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\intertext{and thus that} |
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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) |
|
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\end{align*} |
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as required (c.f. \cite[p. 6]{MR1854636}). |
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\todo{then the general case.} |
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We won't describe a reverse construction (producing a topological $A_\infty$ category |
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from a ``conventional" $A_\infty$ category), but we presume that this will be easy for the experts. |
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} %%%%% end \noop %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |