33 Generally, these sets are indexed by instances of a certain typical shape. |
33 Generally, these sets are indexed by instances of a certain typical shape. |
34 Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, and so on). |
34 Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, and so on). |
35 Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, |
35 Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, |
36 a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, |
36 a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, |
37 and so on. |
37 and so on. |
38 (This allows for strict associativity.) |
38 (This allows for strict associativity; see \cite{ulrike-tillmann-2008,0909.2212}.) |
39 Still other definitions (see, for example, \cite{MR2094071}) |
39 Still other definitions (see, for example, \cite{MR2094071}) |
40 model the $k$-morphisms on more complicated combinatorial polyhedra. |
40 model the $k$-morphisms on more complicated combinatorial polyhedra. |
41 |
41 |
42 For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball. |
42 For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball. |
43 Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic |
43 Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic |