258 In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a |
258 In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a |
259 {\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$. |
259 {\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$. |
260 Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
260 Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
261 We wish to imitate this strategy in higher categories. |
261 We wish to imitate this strategy in higher categories. |
262 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
262 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
263 a product of $k$ intervals \nn{cf xxxx} but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic |
263 a product of $k$ intervals (c.f. \cite{0909.2212}) but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic |
264 to the standard $k$-ball $B^k$. |
264 to the standard $k$-ball $B^k$. |
265 \nn{maybe add that in addition we want functoriality} |
265 \nn{maybe add that in addition we want functoriality} |
266 |
266 |
267 We haven't said precisely what sort of balls we are considering, |
267 We haven't said precisely what sort of balls we are considering, |
268 because we prefer to let this detail be a parameter in the definition. |
268 because we prefer to let this detail be a parameter in the definition. |