205 The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. |
205 The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. |
206 %\nn{we might want a more official looking proof...} |
206 %\nn{we might want a more official looking proof...} |
207 |
207 |
208 We do not insist on surjectivity of the gluing map, since this is not satisfied by all of the examples |
208 We do not insist on surjectivity of the gluing map, since this is not satisfied by all of the examples |
209 we are trying to axiomatize. |
209 we are trying to axiomatize. |
210 If our $k$-morphisms $\cC(X)$ are labeled cell complexes embedded in $X$, then a $k$-morphism is |
210 If our $k$-morphisms $\cC(X)$ are labeled cell complexes embedded in $X$ (c.f. Example \ref{ex:traditional-n-categories} below), then a $k$-morphism is |
211 in the image of the gluing map precisely which the cell complex is in general position |
211 in the image of the gluing map precisely which the cell complex is in general position |
212 with respect to $E$. |
212 with respect to $E$. On the other hand, in categories based on maps to a target space (c.f. Example \ref{ex:maps-to-a-space} below) the gluing map is always surjective |
213 |
213 |
214 If $S$ is a 0-sphere (the case $k=1$ above), then $S$ can be identified with the {\it disjoint} union |
214 If $S$ is a 0-sphere (the case $k=1$ above), then $S$ can be identified with the {\it disjoint} union |
215 of two 0-balls $B_1$ and $B_2$ and the colimit construction $\cl{\cC}(S)$ can be identified |
215 of two 0-balls $B_1$ and $B_2$ and the colimit construction $\cl{\cC}(S)$ can be identified |
216 with the (ordinary, not fibered) product $\cC(B_1) \times \cC(B_2)$. |
216 with the (ordinary, not fibered) product $\cC(B_1) \times \cC(B_2)$. |
217 |
217 |
218 Let $\cl{\cC}(S)\trans E$ denote the image of $\gl_E$. |
218 Let $\cl{\cC}(S)\trans E$ denote the image of $\gl_E$. |
219 We will refer to elements of $\cl{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$". |
219 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. |
220 |
220 |
221 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
221 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
222 as above, then we define $\cC(X)\trans E = \bd^{-1}(\cl{\cC}(\bd X)\trans E)$. |
222 as above, then we define $\cC(X)\trans E = \bd^{-1}(\cl{\cC}(\bd X)\trans E)$. |
223 |
223 |
224 We will call the projection $\cl{\cC}(S)\trans E \to \cC(B_i)$ |
224 We will call the projection $\cl{\cC}(S)\trans E \to \cC(B_i)$ |