21 results that we have avoided here. |
21 results that we have avoided here. |
22 |
22 |
23 \medskip |
23 \medskip |
24 |
24 |
25 Consider first ordinary $n$-categories. |
25 Consider first ordinary $n$-categories. |
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26 \nn{Actually, we're doing both plain and infinity cases here} |
26 We need a set (or sets) of $k$-morphisms for each $0\le k \le n$. |
27 We need a set (or sets) of $k$-morphisms for each $0\le k \le n$. |
27 We must decide on the ``shape" of the $k$-morphisms. |
28 We must decide on the ``shape" of the $k$-morphisms. |
28 Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, ...). |
29 Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, ...). |
29 Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, |
30 Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, |
30 a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, |
31 a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, |
64 \end{axiom} |
65 \end{axiom} |
65 |
66 |
66 |
67 |
67 (Note: We usually omit the subscript $k$.) |
68 (Note: We usually omit the subscript $k$.) |
68 |
69 |
69 We are so far being deliberately vague about what flavor of manifolds we are considering. |
70 We are so far being deliberately vague about what flavor of $k$-balls |
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71 we are considering. |
70 They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. |
72 They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. |
71 They could be topological or PL or smooth. |
73 They could be topological or PL or smooth. |
72 \nn{need to check whether this makes much difference --- see pseudo-isotopy below} |
74 %\nn{need to check whether this makes much difference} |
73 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
75 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
74 to be fussier about corners.) |
76 to be fussier about corners.) |
75 For each flavor of manifold there is a corresponding flavor of $n$-category. |
77 For each flavor of manifold there is a corresponding flavor of $n$-category. |
76 We will concentrate of the case of PL unoriented manifolds. |
78 We will concentrate on the case of PL unoriented manifolds. |
77 |
79 |
78 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
80 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
79 of morphisms). |
81 of morphisms). |
80 The 0-sphere is unusual among spheres in that it is disconnected. |
82 The 0-sphere is unusual among spheres in that it is disconnected. |
81 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
83 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
93 homeomorphisms to the category of sets and bijections. |
95 homeomorphisms to the category of sets and bijections. |
94 \end{axiom} |
96 \end{axiom} |
95 |
97 |
96 (In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries.) |
98 (In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries.) |
97 |
99 |
98 \begin{axiom}[Boundaries (maps)] |
100 \begin{axiom}[Boundaries (maps)]\label{nca-boundary} |
99 For each $k$-ball $X$, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$. |
101 For each $k$-ball $X$, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$. |
100 These maps, for various $X$, comprise a natural transformation of functors. |
102 These maps, for various $X$, comprise a natural transformation of functors. |
101 \end{axiom} |
103 \end{axiom} |
102 |
104 |
103 (Note that the first ``$\bd$" above is part of the data for the category, |
105 (Note that the first ``$\bd$" above is part of the data for the category, |
159 \node[left] at (-1,1) {$B_1$}; |
161 \node[left] at (-1,1) {$B_1$}; |
160 \node[right] at (1,1) {$B_2$}; |
162 \node[right] at (1,1) {$B_2$}; |
161 \end{tikzpicture} |
163 \end{tikzpicture} |
162 $$ |
164 $$ |
163 $$\mathfig{.4}{tempkw/blah3}$$ |
165 $$\mathfig{.4}{tempkw/blah3}$$ |
164 \caption{Combining two balls to get a full boundary}\label{blah3}\end{figure} |
166 \caption{Combining two balls to get a full boundary |
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167 \nn{maybe smaller dots for $E$ in pdf fig}}\label{blah3}\end{figure} |
165 |
168 |
166 Note that we insist on injectivity above. |
169 Note that we insist on injectivity above. |
167 |
170 |
168 Let $\cC(S)_E$ denote the image of $\gl_E$. |
171 Let $\cC(S)_E$ denote the image of $\gl_E$. |
169 We will refer to elements of $\cC(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
172 We will refer to elements of $\cC(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
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173 |
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174 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
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175 as above, then we define $\cC(X)_E = \bd^{-1}(\cC(\bd X)_E)$. |
170 |
176 |
171 We will call the projection $\cC(S)_E \to \cC(B_i)$ |
177 We will call the projection $\cC(S)_E \to \cC(B_i)$ |
172 a {\it restriction} map and write $\res_{B_i}(a)$ |
178 a {\it restriction} map and write $\res_{B_i}(a)$ |
173 (or simply $\res(a)$ when there is no ambiguity), for $a\in \cC(S)_E$. |
179 (or simply $\res(a)$ when there is no ambiguity), for $a\in \cC(S)_E$. |
174 These restriction maps can be thought of as |
180 More generally, we also include under the rubric ``restriction map" the |
175 domain and range maps, relative to the choice of splitting $S = B_1 \cup_E B_2$. |
181 the boundary maps of Axiom \ref{nca-boundary} above, |
176 |
182 another calss of maps introduced after Axion \ref{nca-assoc} below, as well as any composition |
177 If $B$ is a $k$-ball and $E \sub \bd B$ splits $\bd B$ into two $k{-}1$-balls |
183 of restriction maps (inductive definition). |
178 as above, then we define $\cC(B)_E = \bd^{-1}(\cC(\bd B)_E)$. |
184 In particular, we have restriction maps $\cC(X)_E \to \cC(B_i)$ |
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185 ($i = 1, 2$, notation from previous paragraph). |
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186 These restriction maps can be thought of as |
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187 domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. |
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188 |
179 |
189 |
180 Next we consider composition of morphisms. |
190 Next we consider composition of morphisms. |
181 For $n$-categories which lack strong duality, one usually considers |
191 For $n$-categories which lack strong duality, one usually considers |
182 $k$ different types of composition of $k$-morphisms, each associated to a different direction. |
192 $k$ different types of composition of $k$-morphisms, each associated to a different direction. |
183 (For example, vertical and horizontal composition of 2-morphisms.) |
193 (For example, vertical and horizontal composition of 2-morphisms.) |
215 |
225 |
216 \nn{figure \ref{blah6} (blah6) needs a dotted line in the south split ball} |
226 \nn{figure \ref{blah6} (blah6) needs a dotted line in the south split ball} |
217 |
227 |
218 Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$. |
228 Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$. |
219 In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
229 In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
220 a {\it restriction} map and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
230 a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
221 Compositions of boundary and restriction maps will also be called restriction maps. |
231 %Compositions of boundary and restriction maps will also be called restriction maps. |
222 For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
232 %For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
223 restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
233 %restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
224 |
234 |
225 We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$. |
235 We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$. |
226 We will call $\cC(B)_Y$ morphisms which are splittable along $Y$ or transverse to $Y$. |
236 We will call $\cC(B)_Y$ morphisms which are splittable along $Y$ or transverse to $Y$. |
227 We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
237 We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
228 |
238 |