Neurology Mathematics Ideas121.pdf - i i for the parallel transport equations on \u03a3n,\u03ba As a consequence we also have G(n,\u03ba(z = HomT(S(i n,\u03ba i i i(\u2212

Neurology Mathematics Ideas121.pdf - i i for the parallel...

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for the parallel transport equations on Σ ( i ) n,κ . As a consequence we also have G ( i ) n,κ ( z ) = Hom T( S ( i ) n,κ ) ( ,H ( i ) n,κ ( z )), and thus G ( i ) n,κ = H ( i ) n,κ , as claimed. Further, the explicit form of the functor ρ in (B.41) (applied here to the case that M = I ) tells us that G ( i ) n,κ can also be seen as coming from the equivalence M κ I −→ M , with M the bimodule labeling the defect point that is adjacent to the one labeled by I on S n,κ ; thus in particular G ( i ) n,κ is an equivalence. The case ǫ i = 1 is treated analogously. (ii) For analyzing the functor tildewide G ( i ) n,κ in the case ǫ i = 1, we consider an object of the form z = z ǫ 1 1 · · · z ǫ i 1 i 1 z ǫ i +1 i +1 · · · z ǫ n n T( S ( i ) n,κ ). It is straightforward to see that via the balancing of the comonad Z [ κ ] the object tildewide G ( i ) n,κ ( z ) has a canonical structure of an object in T( S n,κ ). The pre-block functor for tildewide Σ ( i ) n,κ takes the values T pre ( tildewide Σ ( i ) n,κ )( x,z ( Q )) = Hom U( S n,κ ) ( U ( x ) ,z
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