106 |
106 |
107 |
107 |
108 \subsection{Plain 2-categories} |
108 \subsection{Plain 2-categories} |
109 \label{ssec:2-cats} |
109 \label{ssec:2-cats} |
110 Let $\cC$ be a topological 2-category. |
110 Let $\cC$ be a topological 2-category. |
111 We will construct a traditional pivotal 2-category. |
111 We will construct from $\cC$ a traditional pivotal 2-category. |
112 (The ``pivotal" corresponds to our assumption of strong duality for $\cC$.) |
112 (The ``pivotal" corresponds to our assumption of strong duality for $\cC$.) |
113 |
113 |
114 We will try to describe the construction in such a way the the generalization to $n>2$ is clear, |
114 We will try to describe the construction in such a way the the generalization to $n>2$ is clear, |
115 though this will make the $n=2$ case a little more complicated than necessary. |
115 though this will make the $n=2$ case a little more complicated than necessary. |
116 |
116 |
117 \nn{Note: We have to decide whether our 2-morphsism are shaped like rectangles or bigons. |
117 Before proceeding, we must decide whether the 2-morphisms of our |
|
118 pivotal 2-category are shaped like rectangles or bigons. |
118 Each approach has advantages and disadvantages. |
119 Each approach has advantages and disadvantages. |
119 For better or worse, we choose bigons here.} |
120 For better or worse, we choose bigons here. |
120 |
|
121 \nn{maybe we should do both rectangles and bigons?} |
|
122 |
121 |
123 Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k)_E$, where $B^k$ denotes the standard |
122 Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k)_E$, where $B^k$ denotes the standard |
124 $k$-ball, which we also think of as the standard bihedron. |
123 $k$-ball, which we also think of as the standard bihedron (a.k.a.\ globe). |
|
124 (For $k=1$ this is an interval, and for $k=2$ it is a bigon.) |
125 Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$ |
125 Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$ |
126 into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$. |
126 into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$. |
127 Recall that the subscript in $\cC(B^k)_E$ means that we consider the subset of $\cC(B^k)$ |
127 Recall that the subscript in $\cC(B^k)_E$ means that we consider the subset of $\cC(B^k)$ |
128 whose boundary is splittable along $E$. |
128 whose boundary is splittable along $E$. |
129 This allows us to define the domain and range of morphisms of $C$ using |
129 This allows us to define the domain and range of morphisms of $C$ using |
134 |
134 |
135 Choosing a homeomorphism $B^2\cup B^2 \to B^2$ defines a ``vertical" composition map |
135 Choosing a homeomorphism $B^2\cup B^2 \to B^2$ defines a ``vertical" composition map |
136 on $C^2$ (Figure \ref{fzo1}). |
136 on $C^2$ (Figure \ref{fzo1}). |
137 Isotopy invariance implies that this is associative. |
137 Isotopy invariance implies that this is associative. |
138 We will define a ``horizontal" composition later. |
138 We will define a ``horizontal" composition later. |
139 \nn{maybe no need to postpone?} |
|
140 |
139 |
141 \begin{figure}[t] |
140 \begin{figure}[t] |
142 \begin{equation*} |
141 \begin{equation*} |
143 \mathfig{.73}{tempkw/zo1} |
142 \mathfig{.73}{tempkw/zo1} |
144 \end{equation*} |
143 \end{equation*} |
145 \caption{Vertical composition of 2-morphisms} |
144 \caption{Vertical composition of 2-morphisms} |
146 \label{fzo1} |
145 \label{fzo1} |
147 \end{figure} |
146 \end{figure} |
148 |
147 |
149 Given $a\in C^1$, define $\id_a = a\times I \in C^1$ (pinched boundary). |
148 Given $a\in C^1$, define $\id_a = a\times I \in C^2$ (pinched boundary). |
150 Extended isotopy invariance for $\cC$ shows that this morphism is an identity for |
149 Extended isotopy invariance for $\cC$ shows that this morphism is an identity for |
151 vertical composition. |
150 vertical composition. |
152 |
151 |
153 Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$. |
152 Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$. |
154 We will show that this 1-morphism is a weak identity. |
153 We will show that this 1-morphism is a weak identity. |
155 This would be easier if our 2-morphisms were shaped like rectangles rather than bigons. |
154 This would be easier if our 2-morphisms were shaped like rectangles rather than bigons. |
|
155 |
|
156 In showing that identity 1-morphisms have the desired properties, we will |
|
157 rely heavily on the extended isotopy invariance of 2-morphisms in $\cC$. |
|
158 This means we are free to add or delete product regions from 2-morphisms. |
|
159 |
156 Let $a: y\to x$ be a 1-morphism. |
160 Let $a: y\to x$ be a 1-morphism. |
157 Define maps $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ |
161 Define 2-morphsims $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ |
158 as shown in Figure \ref{fzo2}. |
162 as shown in Figure \ref{fzo2}. |
159 \begin{figure}[t] |
163 \begin{figure}[t] |
160 \begin{equation*} |
164 \begin{equation*} |
161 \mathfig{.73}{tempkw/zo2} |
165 \mathfig{.73}{tempkw/zo2} |
162 \end{equation*} |
166 \end{equation*} |
163 \caption{blah blah} |
167 \caption{blah blah} |
164 \label{fzo2} |
168 \label{fzo2} |
165 \end{figure} |
169 \end{figure} |
166 In that figure, the red cross-hatched areas are the product of $x$ and a smaller bigon, |
170 As suggested by the figure, these are two different reparameterizations |
167 while the remainder is a half-pinched version of $a\times I$. |
171 of a half-pinched version of $a\times I$. |
168 \nn{the red region is unnecessary; remove it? or does it help? |
|
169 (because it's what you get if you bigonify the natural rectangular picture)} |
|
170 We must show that the two compositions of these two maps give the identity 2-morphisms |
172 We must show that the two compositions of these two maps give the identity 2-morphisms |
171 on $a$ and $a\bullet \id_x$, as defined above. |
173 on $a$ and $a\bullet \id_x$, as defined above. |
172 Figure \ref{fzo3} shows one case. |
174 Figure \ref{fzo3} shows one case. |
173 \begin{figure}[t] |
175 \begin{figure}[t] |
174 \begin{equation*} |
176 \begin{equation*} |
175 \mathfig{.83}{tempkw/zo3} |
177 \mathfig{.83}{tempkw/zo3} |
176 \end{equation*} |
178 \end{equation*} |
177 \caption{blah blah} |
179 \caption{blah blah} |
178 \label{fzo3} |
180 \label{fzo3} |
179 \end{figure} |
181 \end{figure} |
180 In the first step we have inserted a copy of $id(id(x))$ \nn{need better notation for this}. |
182 In the first step we have inserted a copy of $(x\times I)\times I$. |
181 \nn{also need to talk about (somewhere above) |
|
182 how this sort of insertion is allowed by extended isotopy invariance and gluing. |
|
183 Also: maybe half-pinched and unpinched products can be derived from fully pinched |
|
184 products after all (?)} |
|
185 Figure \ref{fzo4} shows the other case. |
183 Figure \ref{fzo4} shows the other case. |
186 \begin{figure}[t] |
184 \begin{figure}[t] |
187 \begin{equation*} |
185 \begin{equation*} |
188 \mathfig{.83}{tempkw/zo4} |
186 \mathfig{.83}{tempkw/zo4} |
189 \end{equation*} |
187 \end{equation*} |
190 \caption{blah blah} |
188 \caption{blah blah} |
191 \label{fzo4} |
189 \label{fzo4} |
192 \end{figure} |
190 \end{figure} |
193 We first collapse the red region, then remove a product morphism from the boundary, |
191 We identify a product region and remove it. |
194 |
192 |
195 We define horizontal composition of 2-morphisms as shown in Figure \ref{fzo5}. |
193 We define horizontal composition of 2-morphisms as shown in Figure \ref{fzo5}. |
196 It is not hard to show that this is independent of the arbitrary (left/right) |
194 It is not hard to show that this is independent of the arbitrary (left/right) |
197 choice made in the definition, and that it is associative. |
195 choice made in the definition, and that it is associative. |
198 \begin{figure}[t] |
196 \begin{figure}[t] |