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 $k$ intervals \nn{cf xxxx} but rather with any $k$-ball, that is, any $k$-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 $k$-ball $B^k$. |
214 to the standard $k$-ball $B^k$. |
215 \nn{maybe add that in addition we want functoriality} |
215 \nn{maybe add that in addition we want functoriality} |
216 |
216 |
217 \nn{say something about different flavors of balls; say it here? later?} |
217 In fact, the axioms here may easily be varied by considering balls with structure (e.g. $m$ independent vector fields, a map to some target space, etc.). Such variations are useful for axiomatizing categories with less duality, and also as technical tools in proofs. |
218 |
218 |
219 \begin{axiom}[Morphisms] |
219 \begin{axiom}[Morphisms] |
220 \label{axiom:morphisms} |
220 \label{axiom:morphisms} |
221 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
221 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
222 the category of $k$-balls and |
222 the category of $k$-balls and |
223 homeomorphisms to the category of sets and bijections. |
223 homeomorphisms to the category of sets and bijections. |
224 \end{axiom} |
224 \end{axiom} |
225 |
225 |
226 Note that the functoriality in the above axiom allows us to operate via \nn{fragment?} |
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227 |
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228 Next we consider domains and ranges of $k$-morphisms. |
226 Next we consider domains and ranges of $k$-morphisms. |
229 Because we assume strong duality, it doesn't make much sense to subdivide the boundary of a morphism |
227 Because we assume strong duality, it doesn't make much sense to subdivide the boundary of a morphism |
230 into domain and range --- the duality operations can convert domain to range and vice-versa. |
228 into domain and range --- the duality operations can convert domain to range and vice-versa. |
231 Instead, we will use a unified domain/range, which we will call a ``boundary". |
229 Instead, we will use a unified domain/range, which we will call a ``boundary". |
232 |
230 |
233 In order to state the axiom for boundaries, we need to extend the functors $\cC_k$ |
231 Later \todo{} we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ from arbitrary manifolds to sets. We need the restriction of these functors to $k$-spheres, for $k<n$, for the next axiom. |
234 of $k$-balls to functors $\cl{\cC}_{k-1}$ of $k$-spheres. |
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235 This extension is described in xxxx below. |
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236 |
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237 %\begin{lem} |
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238 %\label{lem:spheres} |
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239 %For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from |
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240 %the category of $k{-}1$-spheres and |
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241 %homeomorphisms to the category of sets and bijections. |
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242 %\end{lem} |
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243 |
232 |
244 \begin{axiom}[Boundaries]\label{nca-boundary} |
233 \begin{axiom}[Boundaries]\label{nca-boundary} |
245 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 $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
246 These maps, for various $X$, comprise a natural transformation of functors. |
235 These maps, for various $X$, comprise a natural transformation of functors. |
247 \end{axiom} |
236 \end{axiom} |
248 |
237 |
249 \begin{lem}[Boundary from domain and range] |
238 Given two hemispheres (a `domain' and `range') that agree on the equator, we need to be able to assemble them into a boundary value of the entire sphere. |
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239 |
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240 \begin{lem} |
250 \label{lem:domain-and-range} |
241 \label{lem:domain-and-range} |
251 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
242 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
252 $B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
243 $B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
253 Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
244 Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
254 two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$. |
245 two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$. |
255 Then we have an injective map |
246 Then we have an injective map |
256 \[ |
247 \[ |
257 \gl_E : \cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2) \into \cl{\cC}(S) |
248 \gl_E : \cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2) \into \cl{\cC}(S) |
258 \] |
249 \] |
259 which is natural with respect to the actions of homeomorphisms. |
250 which is natural with respect to the actions of homeomorphisms. |
260 (When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product |
251 %(When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product |
261 becomes a normal product.) |
252 %becomes a normal product.) |
262 \end{lem} |
253 \end{lem} |
263 |
254 |
264 \begin{axiom}[Composition] |
255 If $\bdy B = S$, we denote $\bdy^{-1}(\im(\gl_E))$ by $\cC(B)_E$. |
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256 |
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257 \begin{axiom}[Gluing] |
265 \label{axiom:composition} |
258 \label{axiom:composition} |
266 Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
259 Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
267 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
260 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
268 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
261 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
269 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
262 %Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
270 We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
263 We have restriction maps $\cC(B_i)_E \to \cC(Y)$. |
271 Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
264 Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
272 We have a map |
265 We have a map |
273 \[ |
266 \[ |
274 \gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
267 \gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
275 \] |
268 \] |
280 we require that $\gl_Y$ is injective. |
273 we require that $\gl_Y$ is injective. |
281 (For $k=n$ in the plain (non-$A_\infty$) case, see below.) |
274 (For $k=n$ in the plain (non-$A_\infty$) case, see below.) |
282 \end{axiom} |
275 \end{axiom} |
283 |
276 |
284 \begin{axiom}[Strict associativity] \label{nca-assoc} |
277 \begin{axiom}[Strict associativity] \label{nca-assoc} |
285 The composition (gluing) maps above are strictly associative. |
278 The gluing maps above are strictly associative. |
286 Given any decomposition of a ball $B$ into smaller balls |
279 Given any decomposition of a ball $B$ into smaller balls |
287 $$\bigsqcup B_i \to B,$$ |
280 $$\bigsqcup B_i \to B,$$ |
288 any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result. |
281 any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result. |
289 \end{axiom} |
282 \end{axiom} |
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283 For the next axiom, a \emph{pinched product} is a map locally modeled on a degeneracy map between simplices. |
290 \begin{axiom}[Product (identity) morphisms] |
284 \begin{axiom}[Product (identity) morphisms] |
291 \label{axiom:product} |
285 \label{axiom:product} |
292 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
286 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
293 there is a map $\pi^*:\cC(X)\to \cC(E)$. |
287 there is a map $\pi^*:\cC(X)\to \cC(E)$. |
294 These maps must satisfy the following conditions. |
288 These maps must satisfy the following conditions. |