1 %!TEX root = ../blob1.tex

     2 

     3 \section{Comparing $n$-category definitions}

     4 \label{sec:comparing-defs}

     5 

     6 In this appendix we relate the ``topological" category definitions of Section \ref{sec:ncats}

     7 to more traditional definitions, for $n=1$ and 2.

     8 

     9 \subsection{Plain 1-categories}

    10 

    11 Given a topological 1-category $\cC$, we construct a traditional 1-category $C$.

    12 (This is quite straightforward, but we include the details for the sake of completeness and

    13 to shed some light on the $n=2$ case.)

    14 

    15 Let the objects of $C$ be $C^0 \deq \cC(B^0)$ and the morphisms of $C$ be $C^1 \deq \cC(B^1)$, 

    16 where $B^k$ denotes the standard $k$-ball.

    17 The boundary and restriction maps of $\cC$ give domain and range maps from $C^1$ to $C^0$.

    18 

    19 Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$.

    20 Define composition in $C$ to be the induced map $C^1\times C^1 \to C^1$ (defined only when range and domain agree).

    21 By isotopy invariance in $\cC$, any other choice of homeomorphism gives the same composition rule.

    22 Also by isotopy invariance, composition is associative.

    23 

    24 Given $a\in C^0$, define $\id_a \deq a\times B^1$.

    25 By extended isotopy invariance in $\cC$, this has the expected properties of an identity morphism.

    26 

    27 \nn{(slash)id seems to rendering a a boldface 1 --- is this what we want?}

    28 

    29 \medskip

    30 

    31 For 1-categories based on oriented manifolds, there is no additional structure.

    32 

    33 For 1-categories based on unoriented manifolds, there is a map $*:C^1\to C^1$

    34 coming from $\cC$ applied to an orientation-reversing homeomorphism (unique up to isotopy) 

    35 from $B^1$ to itself.

    36 Topological properties of this homeomorphism imply that 

    37 $a^{**} = a$ (* is order 2), * reverses domain and range, and $(ab)^* = b^*a^*$

    38 (* is an anti-automorphism).

    39 

    40 For 1-categories based on Spin manifolds,

    41 the the nontrivial spin homeomorphism from $B^1$ to itself which covers the identity

    42 gives an order 2 automorphism of $C^1$.

    43 

    44 For 1-categories based on $\text{Pin}_-$ manifolds,

    45 we have an order 4 antiautomorphism of $C^1$.

    46 

    47 For 1-categories based on $\text{Pin}_+$ manifolds,

    48 we have an order 2 antiautomorphism and also an order 2 automorphism of $C^1$,

    49 and these two maps commute with each other.

    50 

    51 \nn{need to also consider automorphisms of $B^0$ / objects}

    52 

    53 \medskip

    54 

    55 In the other direction, given a traditional 1-category $C$

    56 (with objects $C^0$ and morphisms $C^1$) we will construct a topological

    57 1-category $\cC$.

    58 

    59 If $X$ is a 0-ball (point), let $\cC(X) \deq C^0$.

    60 If $S$ is a 0-sphere, let $\cC(S) \deq C^0\times C^0$.

    61 If $X$ is a 1-ball, let $\cC(X) \deq C^1$.

    62 Homeomorphisms isotopic to the identity act trivially.

    63 If $C$ has extra structure (e.g.\ it's a *-1-category), we use this structure

    64 to define the action of homeomorphisms not isotopic to the identity

    65 (and get, e.g., an unoriented topological 1-category).

    66 

    67 The domain and range maps of $C$ determine the boundary and restriction maps of $\cC$.

    68 

    69 Gluing maps for $\cC$ are determined my composition of morphisms in $C$.

    70 

    71 For $X$ a 0-ball, $D$ a 1-ball and $a\in \cC(X)$, define the product morphism 

    72 $a\times D \deq \id_a$.

    73 It is not hard to verify that this has the desired properties.

    74 

    75 \medskip

    76 

    77 The compositions of the above two ``arrows" ($\cC\to C\to \cC$ and $C\to \cC\to C$) give back 

    78 more or less exactly the same thing we started with.  

    79 \nn{need better notation here}

    80 As we will see below, for $n>1$ the compositions yield a weaker sort of equivalence.

    81 

    82 \medskip

    83 

    84 Similar arguments show that modules for topological 1-categories are essentially

    85 the same thing as traditional modules for traditional 1-categories.

    86 

    87 \subsection{Plain 2-categories}

    88 

    89 Let $\cC$ be a topological 2-category.

    90 We will construct a traditional pivotal 2-category.

    91 (The ``pivotal" corresponds to our assumption of strong duality for $\cC$.)

    92 

    93 We will try to describe the construction in such a way the the generalization to $n>2$ is clear,

    94 though this will make the $n=2$ case a little more complicated than necessary.

    95 

    96 \nn{Note: We have to decide whether our 2-morphsism are shaped like rectangles or bigons.

    97 Each approach has advantages and disadvantages.

    98 For better or worse, we choose bigons here.}

    99 

   100 \nn{maybe we should do both rectangles and bigons?}

   101 

   102 Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k)_E$, where $B^k$ denotes the standard

   103 $k$-ball, which we also think of as the standard bihedron.

   104 Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$

   105 into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$.

   106 Recall that the subscript in $\cC(B^k)_E$ means that we consider the subset of $\cC(B^k)$

   107 whose boundary is splittable along $E$.

   108 This allows us to define the domain and range of morphisms of $C$ using

   109 boundary and restriction maps of $\cC$.

   110 

   111 Choosing a homeomorphism $B^1\cup B^1 \to B^1$ defines a composition map on $C^1$.

   112 This is not associative, but we will see later that it is weakly associative.

   113 

   114 Choosing a homeomorphism $B^2\cup B^2 \to B^2$ defines a ``vertical" composition map 

   115 on $C^2$ (Figure \ref{fzo1}).

   116 Isotopy invariance implies that this is associative.

   117 We will define a ``horizontal" composition later.

   118 \nn{maybe no need to postpone?}

   119 

   120 \begin{figure}[t]

   121 \begin{equation*}

   122 \mathfig{.73}{tempkw/zo1}

   123 \end{equation*}

   124 \caption{Vertical composition of 2-morphisms}

   125 \label{fzo1}

   126 \end{figure}

   127 

   128 Given $a\in C^1$, define $\id_a = a\times I \in C^1$ (pinched boundary).

   129 Extended isotopy invariance for $\cC$ shows that this morphism is an identity for 

   130 vertical composition.

   131 

   132 Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$.

   133 We will show that this 1-morphism is a weak identity.

   134 This would be easier if our 2-morphisms were shaped like rectangles rather than bigons.

   135 Define let $a: y\to x$ be a 1-morphism.

   136 Define maps $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$

   137 as shown in Figure \ref{fzo2}.

   138 \begin{figure}[t]

   139 \begin{equation*}

   140 \mathfig{.73}{tempkw/zo2}

   141 \end{equation*}

   142 \caption{blah blah}

   143 \label{fzo2}

   144 \end{figure}

   145 In that figure, the red cross-hatched areas are the product of $x$ and a smaller bigon,

   146 while the remained is a half-pinched version of $a\times I$.

   147 \nn{the red region is unnecessary; remove it?  or does it help?

   148 (because it's what you get if you bigonify the natural rectangular picture)}

   149 We must show that the two compositions of these two maps give the identity 2-morphisms

   150 on $a$ and $a\bullet \id_x$, as defined above.

   151 Figure \ref{fzo3} shows one case.

   152 \begin{figure}[t]

   153 \begin{equation*}

   154 \mathfig{.83}{tempkw/zo3}

   155 \end{equation*}

   156 \caption{blah blah}

   157 \label{fzo3}

   158 \end{figure}

   159 In the first step we have inserted a copy of $id(id(x))$ \nn{need better notation for this}.

   160 \nn{also need to talk about (somewhere above) 

   161 how this sort of insertion is allowed by extended isotopy invariance and gluing.

   162 Also: maybe half-pinched and unpinched products can be derived from fully pinched

   163 products after all (?)}

   164 Figure \ref{fzo4} shows the other case.

   165 \begin{figure}[t]

   166 \begin{equation*}

   167 \mathfig{.83}{tempkw/zo4}

   168 \end{equation*}

   169 \caption{blah blah}

   170 \label{fzo4}

   171 \end{figure}

   172 We first collapse the red region, then remove a product morphism from the boundary,

   173 

   174 We define horizontal composition of 2-morphisms as shown in Figure \ref{fzo5}.

   175 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.

   176 \begin{figure}[t]

   177 \begin{equation*}

   178 \mathfig{.83}{tempkw/zo5}

   179 \end{equation*}

   180 \caption{Horizontal composition of 2-morphisms}

   181 \label{fzo5}

   182 \end{figure}

   183 

   184 \nn{need to find a list of axioms for pivotal 2-cats to check}

   185 

   186 \nn{...}

   187 

   188 \medskip

   189 \hrule

   190 \medskip

   191 

   192 \nn{to be continued...}

   193 \medskip