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

   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}

   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}

   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}

   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}

   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}

   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.