208 In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a |
208 In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a |
209 {\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$. |
209 {\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$. |
210 Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
210 Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
211 We wish to imitate this strategy in higher categories. |
211 We wish to imitate this strategy in higher categories. |
212 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
212 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
213 a product of $n$ intervals \nn{cf xxxx} but rather with any $n$-ball, that is, any $n$-manifold which is homeomorphic |
213 a product of $k$ intervals \nn{cf xxxx} but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic |
214 to the standard $n$-ball $B^n$. |
214 to the standard $k$-ball $B^k$. |
215 |
215 \nn{maybe add that in addition we want funtoriality} |
216 \nn{...} |
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217 |
216 |
218 \begin{axiom}[Morphisms] |
217 \begin{axiom}[Morphisms] |
219 \label{axiom:morphisms} |
218 \label{axiom:morphisms} |
220 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
219 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
221 the category of $k$-balls and |
220 the category of $k$-balls and |
222 homeomorphisms to the category of sets and bijections. |
221 homeomorphisms to the category of sets and bijections. |
223 \end{axiom} |
222 \end{axiom} |
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223 |
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224 |
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225 |
224 \begin{lem} |
226 \begin{lem} |
225 \label{lem:spheres} |
227 \label{lem:spheres} |
226 For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from |
228 For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from |
227 the category of $k{-}1$-spheres and |
229 the category of $k{-}1$-spheres and |
228 homeomorphisms to the category of sets and bijections. |
230 homeomorphisms to the category of sets and bijections. |
229 \end{lem} |
231 \end{lem} |
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232 |
230 \begin{axiom}[Boundaries]\label{nca-boundary} |
233 \begin{axiom}[Boundaries]\label{nca-boundary} |
231 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
234 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
232 These maps, for various $X$, comprise a natural transformation of functors. |
235 These maps, for various $X$, comprise a natural transformation of functors. |
233 \end{axiom} |
236 \end{axiom} |
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237 |
234 \begin{lem}[Boundary from domain and range] |
238 \begin{lem}[Boundary from domain and range] |
235 \label{lem:domain-and-range} |
239 \label{lem:domain-and-range} |
236 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
240 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
237 $B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
241 $B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
238 Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
242 Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
243 \] |
247 \] |
244 which is natural with respect to the actions of homeomorphisms. |
248 which is natural with respect to the actions of homeomorphisms. |
245 (When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product |
249 (When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product |
246 becomes a normal product.) |
250 becomes a normal product.) |
247 \end{lem} |
251 \end{lem} |
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252 |
248 \begin{axiom}[Composition] |
253 \begin{axiom}[Composition] |
249 \label{axiom:composition} |
254 \label{axiom:composition} |
250 Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
255 Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
251 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
256 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
252 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
257 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
262 If $k < n$, |
267 If $k < n$, |
263 or if $k=n$ and we are in the $A_\infty$ case, |
268 or if $k=n$ and we are in the $A_\infty$ case, |
264 we require that $\gl_Y$ is injective. |
269 we require that $\gl_Y$ is injective. |
265 (For $k=n$ in the plain (non-$A_\infty$) case, see below.) |
270 (For $k=n$ in the plain (non-$A_\infty$) case, see below.) |
266 \end{axiom} |
271 \end{axiom} |
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272 |
267 \begin{axiom}[Strict associativity] \label{nca-assoc} |
273 \begin{axiom}[Strict associativity] \label{nca-assoc} |
268 The composition (gluing) maps above are strictly associative. |
274 The composition (gluing) maps above are strictly associative. |
269 Given any splitting of a ball $B$ into smaller balls |
275 Given any splitting of a ball $B$ into smaller balls |
270 $$\bigsqcup B_i \to B,$$ |
276 $$\bigsqcup B_i \to B,$$ |
271 any sequence of gluings (in the sense of Definition \ref{defn:gluing-decomposition}, where all the intermediate steps are also disjoint unions of balls) yields the same result. |
277 any sequence of gluings (in the sense of Definition \ref{defn:gluing-decomposition}, where all the intermediate steps are also disjoint unions of balls) yields the same result. |