121 |
121 |
122 |
122 |
123 \subsection{Pivotal 2-categories} |
123 \subsection{Pivotal 2-categories} |
124 \label{ssec:2-cats} |
124 \label{ssec:2-cats} |
125 Let $\cC$ be a disk-like 2-category. |
125 Let $\cC$ be a disk-like 2-category. |
126 We will construct from $\cC$ a traditional pivotal 2-category. |
126 We will construct from $\cC$ a traditional pivotal 2-category $D$. |
127 (The ``pivotal" corresponds to our assumption of strong duality for $\cC$.) |
127 (The ``pivotal" corresponds to our assumption of strong duality for $\cC$.) |
128 |
128 |
129 We will try to describe the construction in such a way that the generalization to $n>2$ is clear, |
129 We will try to describe the construction in such a way that the generalization to $n>2$ is clear, |
130 though this will make the $n=2$ case a little more complicated than necessary. |
130 though this will make the $n=2$ case a little more complicated than necessary. |
131 |
131 |
132 Before proceeding, we must decide whether the 2-morphisms of our |
132 Before proceeding, we must decide whether the 2-morphisms of our |
133 pivotal 2-category are shaped like rectangles or bigons. |
133 pivotal 2-category are shaped like rectangles or bigons. |
134 Each approach has advantages and disadvantages. |
134 Each approach has advantages and disadvantages. |
135 For better or worse, we choose bigons here. |
135 For better or worse, we choose bigons here. |
136 |
136 |
137 Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k) \trans E$, where $B^k$ denotes the standard |
137 Define the $k$-morphisms $D^k$ of $D$ to be $\cC(B^k) \trans E$, where $B^k$ denotes the standard |
138 $k$-ball, which we also think of as the standard bihedron (a.k.a.\ globe). |
138 $k$-ball, which we also think of as the standard bihedron (a.k.a.\ globe). |
139 (For $k=1$ this is an interval, and for $k=2$ it is a bigon.) |
139 (For $k=1$ this is an interval, and for $k=2$ it is a bigon.) |
140 Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$ |
140 Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$ |
141 into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$. |
141 into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$. |
142 Recall that the subscript in $\cC(B^k) \trans E$ means that we consider the subset of $\cC(B^k)$ |
142 Recall that the subscript in $\cC(B^k) \trans E$ means that we consider the subset of $\cC(B^k)$ |
143 whose boundary is splittable along $E$. |
143 whose boundary is splittable along $E$. |
144 This allows us to define the domain and range of morphisms of $C$ using |
144 This allows us to define the domain and range of morphisms of $D$ using |
145 boundary and restriction maps of $\cC$. |
145 boundary and restriction maps of $\cC$. |
146 |
146 |
147 Choosing a homeomorphism $B^1\cup B^1 \to B^1$ defines a composition map on $C^1$. |
147 Choosing a homeomorphism $B^1\cup B^1 \to B^1$ defines a composition map on $D^1$. |
148 This is not associative, but we will see later that it is weakly associative. |
148 This is not associative, but we will see later that it is weakly associative. |
149 |
149 |
150 Choosing a homeomorphism $B^2\cup B^2 \to B^2$ defines a ``vertical" composition map |
150 Choosing a homeomorphism $B^2\cup B^2 \to B^2$ defines a ``vertical" composition map |
151 on $C^2$ (Figure \ref{fzo1}). |
151 on $D^2$ (Figure \ref{fzo1}). |
152 Isotopy invariance implies that this is associative. |
152 Isotopy invariance implies that this is associative. |
153 We will define a ``horizontal" composition later. |
153 We will define a ``horizontal" composition later. |
154 |
154 |
155 \begin{figure}[t] |
155 \begin{figure}[t] |
156 \centering |
156 \centering |
201 \end{tikzpicture} |
201 \end{tikzpicture} |
202 \caption{Vertical composition of 2-morphisms} |
202 \caption{Vertical composition of 2-morphisms} |
203 \label{fzo1} |
203 \label{fzo1} |
204 \end{figure} |
204 \end{figure} |
205 |
205 |
206 Given $a\in C^1$, define $\id_a = a\times I \in C^2$ (pinched boundary). |
206 Given $a\in D^1$, define $\id_a = a\times I \in D^2$ (pinched boundary). |
207 Extended isotopy invariance for $\cC$ shows that this morphism is an identity for |
207 Extended isotopy invariance for $\cC$ shows that this morphism is an identity for |
208 vertical composition. |
208 vertical composition. |
209 |
209 |
210 Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$. |
210 Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$. |
211 We will show that this 1-morphism is a weak identity. |
211 We will show that this 1-morphism is a weak identity. |