212 If $S$ is a 0-sphere (the case $k=1$ above), then $S$ can be identified with the {\it disjoint} union |
212 If $S$ is a 0-sphere (the case $k=1$ above), then $S$ can be identified with the {\it disjoint} union |
213 of two 0-balls $B_1$ and $B_2$ and the colimit construction $\cl{\cC}(S)$ can be identified |
213 of two 0-balls $B_1$ and $B_2$ and the colimit construction $\cl{\cC}(S)$ can be identified |
214 with the (ordinary, not fibered) product $\cC(B_1) \times \cC(B_2)$. |
214 with the (ordinary, not fibered) product $\cC(B_1) \times \cC(B_2)$. |
215 |
215 |
216 Let $\cl{\cC}(S)\trans E$ denote the image of $\gl_E$. |
216 Let $\cl{\cC}(S)\trans E$ denote the image of $\gl_E$. |
217 We will refer to elements of $\cl{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$". When the gluing map is surjective every such element is splittable. |
217 We will refer to elements of $\cl{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$". |
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218 When the gluing map is surjective every such element is splittable. |
218 |
219 |
219 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
220 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
220 as above, then we define $\cC(X)\trans E = \bd^{-1}(\cl{\cC}(\bd X)\trans E)$. |
221 as above, then we define $\cC(X)\trans E = \bd^{-1}(\cl{\cC}(\bd X)\trans E)$. |
221 |
222 |
222 We will call the projection $\cl{\cC}(S)\trans E \to \cC(B_i)$ |
223 We will call the projection $\cl{\cC}(S)\trans E \to \cC(B_i)$ given by the composition |
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224 $$\cl{\cC}(S)\trans E \xrightarrow{\gl^{-1}} \cC(B_1) \times \cC(B_2) \xrightarrow{\pr_i} \cC(B_i)$$ |
223 a {\it restriction} map and write $\res_{B_i}(a)$ |
225 a {\it restriction} map and write $\res_{B_i}(a)$ |
224 (or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)\trans E$. |
226 (or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)\trans E$. |
225 More generally, we also include under the rubric ``restriction map" |
227 More generally, we also include under the rubric ``restriction map" |
226 the boundary maps of Axiom \ref{nca-boundary} above, |
228 the boundary maps of Axiom \ref{nca-boundary} above, |
227 another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition |
229 another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition |
228 of restriction maps. |
230 of restriction maps. |
229 In particular, we have restriction maps $\cC(X)\trans E \to \cC(B_i)$ |
231 In particular, we have restriction maps $\cC(X)\trans E \to \cC(B_i)$ |
230 ($i = 1, 2$, notation from previous paragraph). |
232 defined as the composition of the boundary with the first restriction map described above: |
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233 $$ |
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234 \cC(X) \trans E \xrightarrow{\bdy} \cl{\cC}(\bdy X)\trans E \xrightarrow{\res} \cC(B_i) |
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235 .$$ |
231 These restriction maps can be thought of as |
236 These restriction maps can be thought of as |
232 domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. |
237 domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. |
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238 These restriction maps in fact have their image in the subset $\cC(B_i)\trans E$, |
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239 and so to emphasize this we will sometimes write the restriction map as $\cC(X)\trans E \to \cC(B_i)\trans E$. |
233 |
240 |
234 |
241 |
235 Next we consider composition of morphisms. |
242 Next we consider composition of morphisms. |
236 For $n$-categories which lack strong duality, one usually considers |
243 For $n$-categories which lack strong duality, one usually considers |
237 $k$ different types of composition of $k$-morphisms, each associated to a different ``direction". |
244 $k$ different types of composition of $k$-morphisms, each associated to a different ``direction". |