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202 \caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
202 \caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
203 |
203 |
204 Note that we insist on injectivity above. |
204 Note that we insist on injectivity above. |
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...} |
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207 |
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208 We do not insist on surjectivity of the gluing map, since this is not satisfied by all of the examples |
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209 we are trying to axiomatize. |
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210 If our $k$-morphisms $\cC(X)$ are labeled cell complexes embedded in $X$, then a $k$-morphism is |
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211 in the image of the gluing map precisely which the cell complex is in general position |
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212 with respect to $E$. |
207 |
213 |
208 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 |
209 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 |
210 with the (ordinary, not fibered) product $\cC(B_1) \times \cC(B_2)$. |
216 with the (ordinary, not fibered) product $\cC(B_1) \times \cC(B_2)$. |
211 |
217 |