496 In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a |
496 In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a |
497 {\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$. |
497 {\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$. |
498 Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
498 Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
499 We wish to imitate this strategy in higher categories. |
499 We wish to imitate this strategy in higher categories. |
500 Because we are mainly interested in the case of pivotal $n$-categories, we replace the intervals $[0,r]$ not with |
500 Because we are mainly interested in the case of pivotal $n$-categories, we replace the intervals $[0,r]$ not with |
501 a product of $k$ intervals (c.f. \cite{ulrike-tillmann-2008,0909.2212}) but rather with any $k$-ball, that is, |
501 a product of $k$ intervals (c.f.\ \cite{0909.2212}) but rather with any $k$-ball, that is, |
|
502 % \cite{ulrike-tillmann-2008,0909.2212} |
502 any $k$-manifold which is homeomorphic |
503 any $k$-manifold which is homeomorphic |
503 to the standard $k$-ball $B^k$. |
504 to the standard $k$-ball $B^k$. |
504 |
505 |
505 By default our balls are unoriented, |
506 By default our balls are unoriented, |
506 but it is useful at times to vary this, |
507 but it is useful at times to vary this, |
1291 \newblock {\em Infinite loop spaces}, volume~90 of {\em Annals of Mathematics |
1292 \newblock {\em Infinite loop spaces}, volume~90 of {\em Annals of Mathematics |
1292 Studies}. |
1293 Studies}. |
1293 \newblock Princeton University Press, Princeton, N.J., 1978. |
1294 \newblock Princeton University Press, Princeton, N.J., 1978. |
1294 \newblock \mathscinet{MR505692} \googlebooks{e2rYkg9lGnsC}. |
1295 \newblock \mathscinet{MR505692} \googlebooks{e2rYkg9lGnsC}. |
1295 |
1296 |
1296 \bibitem{ulrike-tillmann-2008} |
1297 %\bibitem{ulrike-tillmann-2008} |
1297 Ulrike Tillmann, 2008. |
1298 %Ulrike Tillmann, 2008. |
1298 \newblock personal communication. |
1299 %\newblock personal communication. |
1299 |
1300 |
1300 \bibitem{0909.2212} |
1301 \bibitem{0909.2212} |
1301 Ronald {Brown}. |
1302 Ronald {Brown}. |
1302 \newblock {Moore hyperrectangles on a space form a strict cubical |
1303 \newblock {Moore hyperrectangles on a space form a strict cubical |
1303 omega-category}, September 2009. |
1304 omega-category}, September 2009. |