equal
deleted
inserted
replaced
3249 shows the intertwiners we need. |
3249 shows the intertwiners we need. |
3250 Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle |
3250 Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle |
3251 on the boundary. |
3251 on the boundary. |
3252 This is the 3-dimensional part of the data for the Morita equivalence. |
3252 This is the 3-dimensional part of the data for the Morita equivalence. |
3253 (Note that, by symmetry, the $c$ and $d$ arrows of Figure \ref{morita-fig-2} |
3253 (Note that, by symmetry, the $c$ and $d$ arrows of Figure \ref{morita-fig-2} |
3254 are the same (up to rotation), as are the $h$ and $g$ arrows.) |
3254 are the same (up to rotation), as the $h$ and $g$ arrows.) |
3255 |
3255 |
3256 In order for these 3-morphisms to be equivalences, |
3256 In order for these 3-morphisms to be equivalences, |
3257 they must be invertible (i.e.\ $a=b\inv$, $c=d\inv$, $e=f\inv$) and in addition |
3257 they must be invertible (i.e.\ $a=b\inv$, $c=d\inv$, $e=f\inv$) and in addition |
3258 they must satisfy identities corresponding to Morse cancellations on 2-manifolds. |
3258 they must satisfy identities corresponding to Morse cancellations on 2-manifolds. |
3259 These are illustrated in Figure \ref{morita-fig-3}. |
3259 These are illustrated in Figure \ref{morita-fig-3}. |