--- a/text/appendixes/comparing_defs.tex Fri Jun 04 20:37:38 2010 -0700
+++ b/text/appendixes/comparing_defs.tex Fri Jun 04 20:43:14 2010 -0700
@@ -12,14 +12,18 @@
\subsection{$1$-categories over $\Set$ or $\Vect$}
\label{ssec:1-cats}
Given a topological $1$-category $\cX$ we construct a $1$-category in the conventional sense, $c(\cX)$.
-This construction is quite straightforward, but we include the details for the sake of completeness, because it illustrates the role of structures (e.g. orientations, spin structures, etc) on the underlying manifolds, and
+This construction is quite straightforward, but we include the details for the sake of completeness,
+because it illustrates the role of structures (e.g. orientations, spin structures, etc)
+on the underlying manifolds, and
to shed some light on the $n=2$ case, which we describe in \S \ref{ssec:2-cats}.
Let $B^k$ denote the \emph{standard} $k$-ball.
-Let the objects of $c(\cX)$ be $c(\cX)^0 = \cX(B^0)$ and the morphisms of $c(\cX)$ be $c(\cX)^1 = \cX(B^1)$. The boundary and restriction maps of $\cX$ give domain and range maps from $c(\cX)^1$ to $c(\cX)^0$.
+Let the objects of $c(\cX)$ be $c(\cX)^0 = \cX(B^0)$ and the morphisms of $c(\cX)$ be $c(\cX)^1 = \cX(B^1)$.
+The boundary and restriction maps of $\cX$ give domain and range maps from $c(\cX)^1$ to $c(\cX)^0$.
Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$.
-Define composition in $c(\cX)$ to be the induced map $c(\cX)^1\times c(\cX)^1 \to c(\cX)^1$ (defined only when range and domain agree).
+Define composition in $c(\cX)$ to be the induced map $c(\cX)^1\times c(\cX)^1 \to c(\cX)^1$
+(defined only when range and domain agree).
By isotopy invariance in $\cX$, any other choice of homeomorphism gives the same composition rule.
Also by isotopy invariance, composition is strictly associative.
@@ -27,9 +31,12 @@
By extended isotopy invariance in $\cX$, this has the expected properties of an identity morphism.
-If the underlying manifolds for $\cX$ have further geometric structure, then we obtain certain functors. The base case is for oriented manifolds, where we obtain no extra algebraic data.
+If the underlying manifolds for $\cX$ have further geometric structure, then we obtain certain functors.
+The base case is for oriented manifolds, where we obtain no extra algebraic data.
-For 1-categories based on unoriented manifolds (somewhat confusingly, we're thinking of being unoriented as requiring extra data beyond being oriented, namely the identification between the orientations), there is a map $*:c(\cX)^1\to c(\cX)^1$
+For 1-categories based on unoriented manifolds (somewhat confusingly, we're thinking of being
+unoriented as requiring extra data beyond being oriented, namely the identification between the orientations),
+there is a map $*:c(\cX)^1\to c(\cX)^1$
coming from $\cX$ applied to an orientation-reversing homeomorphism (unique up to isotopy)
from $B^1$ to itself.
Topological properties of this homeomorphism imply that
@@ -71,7 +78,8 @@
\medskip
-The compositions of the constructions above, $$\cX\to c(\cX)\to t(c(\cX))$$ and $$C\to t(C)\to c(t(C)),$$ give back
+The compositions of the constructions above, $$\cX\to c(\cX)\to t(c(\cX))$$
+and $$C\to t(C)\to c(t(C)),$$ give back
more or less exactly the same thing we started with.
As we will see below, for $n>1$ the compositions yield a weaker sort of equivalence.
@@ -169,7 +177,8 @@
We first collapse the red region, then remove a product morphism from the boundary,
We define horizontal composition of 2-morphisms as shown in Figure \ref{fzo5}.
-It is not hard to show that this is independent of the arbitrary (left/right) choice made in the definition, and that it is associative.
+It is not hard to show that this is independent of the arbitrary (left/right)
+choice made in the definition, and that it is associative.
\begin{figure}[t]
\begin{equation*}
\mathfig{.83}{tempkw/zo5}
@@ -191,22 +200,48 @@
\subsection{$A_\infty$ $1$-categories}
\label{sec:comparing-A-infty}
-In this section, we make contact between the usual definition of an $A_\infty$ algebra and our definition of a topological $A_\infty$ algebra, from Definition \ref{defn:topological-Ainfty-category}.
+In this section, we make contact between the usual definition of an $A_\infty$ algebra
+and our definition of a topological $A_\infty$ algebra, from Definition \ref{defn:topological-Ainfty-category}.
-We begin be restricting the data of a topological $A_\infty$ algebra to the standard interval $[0,1]$, which we can alternatively characterise as:
+We begin be restricting the data of a topological $A_\infty$ algebra to the standard interval $[0,1]$,
+which we can alternatively characterise as:
\begin{defn}
-A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$, and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with
+A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$,
+and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with
\begin{itemize}
\item an action of the operad of $\Obj(\cC)$-labeled cell decompositions
\item and a compatible action of $\CD{[0,1]}$.
\end{itemize}
\end{defn}
-Here the operad of cell decompositions of $[0,1]$ has operations indexed by a finite set of points $0 < x_1< \cdots < x_k < 1$, cutting $[0,1]$ into subintervals. An $X$-labeled cell decomposition labels $\{0, x_1, \ldots, x_k, 1\}$ by $X$. Given two cell decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$, we can compose them to form a new cell decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$ by inserting the points of $\cJ^{(2)}$ linearly into the $m$-th interval of $\cJ^{(1)}$. In the $X$-labeled case, we insist that the appropriate labels match up. Saying we have an action of this operad means that for each labeled cell decomposition $0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain map $$\cC_{a_0,a_1} \tensor \cdots \tensor \cC_{a_k,a_{k+1}} \to \cC(a_0,a_{k+1})$$ and these chain maps compose exactly as the cell decompositions.
-An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which is supported on the subintervals determined by $\pi$, then the two possible operations (glue intervals together, then apply the diffeomorphisms, or apply the diffeormorphisms separately to the subintervals, then glue) commute (as usual, up to a weakly unique homotopy).
+Here the operad of cell decompositions of $[0,1]$ has operations indexed by a finite set of
+points $0 < x_1< \cdots < x_k < 1$, cutting $[0,1]$ into subintervals.
+An $X$-labeled cell decomposition labels $\{0, x_1, \ldots, x_k, 1\}$ by $X$.
+Given two cell decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$, we can compose
+them to form a new cell decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$ by inserting the points
+of $\cJ^{(2)}$ linearly into the $m$-th interval of $\cJ^{(1)}$.
+In the $X$-labeled case, we insist that the appropriate labels match up.
+Saying we have an action of this operad means that for each labeled cell decomposition
+$0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain
+map $$\cC_{a_0,a_1} \tensor \cdots \tensor \cC_{a_k,a_{k+1}} \to \cC(a_0,a_{k+1})$$ and these
+chain maps compose exactly as the cell decompositions.
+An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad
+if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which
+is supported on the subintervals determined by $\pi$, then the two possible operations
+(glue intervals together, then apply the diffeomorphisms, or apply the diffeormorphisms
+separately to the subintervals, then glue) commute (as usual, up to a weakly unique homotopy).
-Translating between this notion and the usual definition of an $A_\infty$ category is now straightforward. To restrict to the standard interval, define $\cC_{a,b} = \cC([0,1];a,b)$. Given a cell decomposition $0 < x_1< \cdots < x_k < 1$, we use the map (suppressing labels)
+Translating between this notion and the usual definition of an $A_\infty$ category is now straightforward.
+To restrict to the standard interval, define $\cC_{a,b} = \cC([0,1];a,b)$.
+Given a cell decomposition $0 < x_1< \cdots < x_k < 1$, we use the map (suppressing labels)
$$\cC([0,1])^{\tensor k+1} \to \cC([0,x_1]) \tensor \cdots \tensor \cC[x_k,1] \to \cC([0,1])$$
-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. The action of $\CD{[0,1]}$ carries across, and is automatically compatible. Going the other way, we just declare $\cC(J;a,b) = \cC_{a,b}$, pick a diffeomorphism $\phi_J : J \isoto [0,1]$ for every interval $J$, define the gluing map $\cC(J_1) \tensor \cC(J_2) \to \cC(J_1 \cup J_2)$ by the first applying the cell decomposition map for $0 < \frac{1}{2} < 1$, then the self-diffeomorphism of $[0,1]$ given by $\frac{1}{2} (\phi_{J_1} \cup (1+ \phi_{J_2})) \circ \phi_{J_1 \cup J_2}^{-1}$. You can readily check that this gluing map is associative on the nose. \todo{really?}
+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.
+The action of $\CD{[0,1]}$ carries across, and is automatically compatible.
+Going the other way, we just declare $\cC(J;a,b) = \cC_{a,b}$, pick a diffeomorphism
+$\phi_J : J \isoto [0,1]$ for every interval $J$, define the gluing map
+$\cC(J_1) \tensor \cC(J_2) \to \cC(J_1 \cup J_2)$ by the first applying
+the cell decomposition map for $0 < \frac{1}{2} < 1$, then the self-diffeomorphism of $[0,1]$
+given by $\frac{1}{2} (\phi_{J_1} \cup (1+ \phi_{J_2})) \circ \phi_{J_1 \cup J_2}^{-1}$.
+You can readily check that this gluing map is associative on the nose. \todo{really?}
%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 $\cJ^{(1)} \circ_m \cJ^{(2)}$.
@@ -235,7 +270,15 @@
%\end{enumerate}
%\end{defn}
-From a topological $A_\infty$ category on $[0,1]$ $\cC$ we can produce a `conventional' $A_\infty$ category $(A, \{m_k\})$ as defined in, for example, \cite{MR1854636}. We'll just describe the algebra case (that is, a category with only one object), as the modifications required to deal with multiple objects are trivial. Define $A = \cC$ as a chain complex (so $m_1 = d$). Define $m_2 : A\tensor A \to A$ by $f_{\{(0,\frac{1}{2}),(\frac{1}{2},1)\}}$. To define $m_3$, we begin by taking the one parameter family $\phi_3$ of diffeomorphisms of $[0,1]$ that interpolates linearly between the identity and the piecewise linear diffeomorphism taking $\frac{1}{4}$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $\frac{3}{4}$, and then define
+From a topological $A_\infty$ category on $[0,1]$ $\cC$ we can produce a `conventional'
+$A_\infty$ category $(A, \{m_k\})$ as defined in, for example, \cite{MR1854636}.
+We'll just describe the algebra case (that is, a category with only one object),
+as the modifications required to deal with multiple objects are trivial.
+Define $A = \cC$ as a chain complex (so $m_1 = d$).
+Define $m_2 : A\tensor A \to A$ by $f_{\{(0,\frac{1}{2}),(\frac{1}{2},1)\}}$.
+To define $m_3$, we begin by taking the one parameter family $\phi_3$ of diffeomorphisms
+of $[0,1]$ that interpolates linearly between the identity and the piecewise linear
+diffeomorphism taking $\frac{1}{4}$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $\frac{3}{4}$, and then define
\begin{equation*}
m_3(a,b,c) = ev(\phi_3, m_2(m_2(a,b), c)).
\end{equation*}
@@ -250,4 +293,5 @@
\end{align*}
as required (c.f. \cite[p. 6]{MR1854636}).
\todo{then the general case.}
-We won't describe a reverse construction (producing a topological $A_\infty$ category from a `conventional' $A_\infty$ category), but we presume that this will be easy for the experts.
\ No newline at end of file
+We won't describe a reverse construction (producing a topological $A_\infty$ category
+from a `conventional' $A_\infty$ category), but we presume that this will be easy for the experts.
\ No newline at end of file