297 In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a |
297 In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a |
298 {\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$. |
298 {\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$. |
299 Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
299 Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
300 We wish to imitate this strategy in higher categories. |
300 We wish to imitate this strategy in higher categories. |
301 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
301 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
302 a product of $k$ intervals (c.f. \cite{0909.2212}) but rather with any $k$-ball, that is, |
302 a product of $k$ intervals (c.f. \cite{ulrike-tillmann-2008,0909.2212}) but rather with any $k$-ball, that is, |
303 any $k$-manifold which is homeomorphic |
303 any $k$-manifold which is homeomorphic |
304 to the standard $k$-ball $B^k$. |
304 to the standard $k$-ball $B^k$. |
305 |
305 |
306 By default our balls are unoriented, |
306 By default our balls are unoriented, |
307 but it is useful at times to vary this, |
307 but it is useful at times to vary this, |