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-%!TEX root = ../blob1.tex
-
-\section{Comparing $n$-category definitions}
-\label{sec:comparing-defs}
-
-In this appendix we relate the ``topological" category definitions of Section \ref{sec:ncats}
-to more traditional definitions, for $n=1$ and 2.
-
-\subsection{Plain 1-categories}
-
-Given a topological 1-category $\cC$, we construct a traditional 1-category $C$.
-(This is quite straightforward, but we include the details for the sake of completeness and
-to shed some light on the $n=2$ case.)
-
-Let the objects of $C$ be $C^0 \deq \cC(B^0)$ and the morphisms of $C$ be $C^1 \deq \cC(B^1)$, 
-where $B^k$ denotes the standard $k$-ball.
-The boundary and restriction maps of $\cC$ give domain and range maps from $C^1$ to $C^0$.
-
-Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$.
-Define composition in $C$ to be the induced map $C^1\times C^1 \to C^1$ (defined only when range and domain agree).
-By isotopy invariance in $\cC$, any other choice of homeomorphism gives the same composition rule.
-Also by isotopy invariance, composition is associative.
-
-Given $a\in C^0$, define $\id_a \deq a\times B^1$.
-By extended isotopy invariance in $\cC$, this has the expected properties of an identity morphism.
-
-\nn{(slash)id seems to rendering a a boldface 1 --- is this what we want?}
-
-\medskip
-
-For 1-categories based on oriented manifolds, there is no additional structure.
-
-For 1-categories based on unoriented manifolds, there is a map $*:C^1\to C^1$
-coming from $\cC$ applied to an orientation-reversing homeomorphism (unique up to isotopy) 
-from $B^1$ to itself.
-Topological properties of this homeomorphism imply that 
-$a^{**} = a$ (* is order 2), * reverses domain and range, and $(ab)^* = b^*a^*$
-(* is an anti-automorphism).
-
-For 1-categories based on Spin manifolds,
-the the nontrivial spin homeomorphism from $B^1$ to itself which covers the identity
-gives an order 2 automorphism of $C^1$.
-
-For 1-categories based on $\text{Pin}_-$ manifolds,
-we have an order 4 antiautomorphism of $C^1$.
-
-For 1-categories based on $\text{Pin}_+$ manifolds,
-we have an order 2 antiautomorphism and also an order 2 automorphism of $C^1$,
-and these two maps commute with each other.
-
-\nn{need to also consider automorphisms of $B^0$ / objects}
-
-\medskip
-
-In the other direction, given a traditional 1-category $C$
-(with objects $C^0$ and morphisms $C^1$) we will construct a topological
-1-category $\cC$.
-
-If $X$ is a 0-ball (point), let $\cC(X) \deq C^0$.
-If $S$ is a 0-sphere, let $\cC(S) \deq C^0\times C^0$.
-If $X$ is a 1-ball, let $\cC(X) \deq C^1$.
-Homeomorphisms isotopic to the identity act trivially.
-If $C$ has extra structure (e.g.\ it's a *-1-category), we use this structure
-to define the action of homeomorphisms not isotopic to the identity
-(and get, e.g., an unoriented topological 1-category).
-
-The domain and range maps of $C$ determine the boundary and restriction maps of $\cC$.
-
-Gluing maps for $\cC$ are determined my composition of morphisms in $C$.
-
-For $X$ a 0-ball, $D$ a 1-ball and $a\in \cC(X)$, define the product morphism 
-$a\times D \deq \id_a$.
-It is not hard to verify that this has the desired properties.
-
-\medskip
-
-The compositions of the above two ``arrows" ($\cC\to C\to \cC$ and $C\to \cC\to C$) give back 
-more or less exactly the same thing we started with.  
-\nn{need better notation here}
-As we will see below, for $n>1$ the compositions yield a weaker sort of equivalence.
-
-\medskip
-
-Similar arguments show that modules for topological 1-categories are essentially
-the same thing as traditional modules for traditional 1-categories.
-
-\subsection{Plain 2-categories}
-
-Let $\cC$ be a topological 2-category.
-We will construct a traditional pivotal 2-category.
-(The ``pivotal" corresponds to our assumption of strong duality for $\cC$.)
-
-We will try to describe the construction in such a way the the generalization to $n>2$ is clear,
-though this will make the $n=2$ case a little more complicated than necessary.
-
-\nn{Note: We have to decide whether our 2-morphsism are shaped like rectangles or bigons.
-Each approach has advantages and disadvantages.
-For better or worse, we choose bigons here.}
-
-\nn{maybe we should do both rectangles and bigons?}
-
-Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k)_E$, where $B^k$ denotes the standard
-$k$-ball, which we also think of as the standard bihedron.
-Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$
-into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$.
-Recall that the subscript in $\cC(B^k)_E$ means that we consider the subset of $\cC(B^k)$
-whose boundary is splittable along $E$.
-This allows us to define the domain and range of morphisms of $C$ using
-boundary and restriction maps of $\cC$.
-
-Choosing a homeomorphism $B^1\cup B^1 \to B^1$ defines a composition map on $C^1$.
-This is not associative, but we will see later that it is weakly associative.
-
-Choosing a homeomorphism $B^2\cup B^2 \to B^2$ defines a ``vertical" composition map 
-on $C^2$ (Figure \ref{fzo1}).
-Isotopy invariance implies that this is associative.
-We will define a ``horizontal" composition later.
-\nn{maybe no need to postpone?}
-
-\begin{figure}[t]
-\begin{equation*}
-\mathfig{.73}{tempkw/zo1}
-\end{equation*}
-\caption{Vertical composition of 2-morphisms}
-\label{fzo1}
-\end{figure}
-
-Given $a\in C^1$, define $\id_a = a\times I \in C^1$ (pinched boundary).
-Extended isotopy invariance for $\cC$ shows that this morphism is an identity for 
-vertical composition.
-
-Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$.
-We will show that this 1-morphism is a weak identity.
-This would be easier if our 2-morphisms were shaped like rectangles rather than bigons.
-Define let $a: y\to x$ be a 1-morphism.
-Define maps $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$
-as shown in Figure \ref{fzo2}.
-\begin{figure}[t]
-\begin{equation*}
-\mathfig{.73}{tempkw/zo2}
-\end{equation*}
-\caption{blah blah}
-\label{fzo2}
-\end{figure}
-In that figure, the red cross-hatched areas are the product of $x$ and a smaller bigon,
-while the remained is a half-pinched version of $a\times I$.
-\nn{the red region is unnecessary; remove it?  or does it help?
-(because it's what you get if you bigonify the natural rectangular picture)}
-We must show that the two compositions of these two maps give the identity 2-morphisms
-on $a$ and $a\bullet \id_x$, as defined above.
-Figure \ref{fzo3} shows one case.
-\begin{figure}[t]
-\begin{equation*}
-\mathfig{.83}{tempkw/zo3}
-\end{equation*}
-\caption{blah blah}
-\label{fzo3}
-\end{figure}
-In the first step we have inserted a copy of $id(id(x))$ \nn{need better notation for this}.
-\nn{also need to talk about (somewhere above) 
-how this sort of insertion is allowed by extended isotopy invariance and gluing.
-Also: maybe half-pinched and unpinched products can be derived from fully pinched
-products after all (?)}
-Figure \ref{fzo4} shows the other case.
-\begin{figure}[t]
-\begin{equation*}
-\mathfig{.83}{tempkw/zo4}
-\end{equation*}
-\caption{blah blah}
-\label{fzo4}
-\end{figure}
-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.
-\begin{figure}[t]
-\begin{equation*}
-\mathfig{.83}{tempkw/zo5}
-\end{equation*}
-\caption{Horizontal composition of 2-morphisms}
-\label{fzo5}
-\end{figure}
-
-\nn{need to find a list of axioms for pivotal 2-cats to check}
-
-\nn{...}
-
-\medskip
-\hrule
-\medskip
-
-\nn{to be continued...}
-\medskip