Neurology Mathematics Ideas120.pdf - i Let us describe the...

Let us describe the functor G
(
i
)
n,κ
in detail. First note that for z
ǫ
1
1
⊠ · · · ⊠ z
ǫ
n
n
∈ T( S
n,κ
) and
ǫ
i
= − 1 the object
Hom
A
( z
i
, 1 ) ⊗ z
ǫ
1
1
⊠ · · ·
hatwider
z
ǫ
i
i
· · · ⊠ z
ǫ
n
n
∈ T( S
(
i
)
n,κ
) (C.7)
(with the symbol hatwide z indicating that the factor z is to be removed from the expression) comes
canonically with the following balancings: For a ∈ A the balancing between z
i − 1
and z
i +1
is, in
case the orientations are as in the picture (C.3),
Hom
A
( z
i
, 1 ) ⊗ z
i − 1
.a ⊠ z
i +1
= Hom
A
( z
i
, 1 ) ⊗ ( z
i − 1
⊗ a ) ⊠ z
i +1
∼
= Hom
A
( z
i
⊗ [ κ
i
] a, 1 ) ⊗ z
i − 1
⊠ z
i +1
∼
= Hom
A
( z
i
, [ κ
i
− 1] a ) ⊗ z
i − 1
⊠ z
i +1
∼
= Hom
A
( [ κ
i
− 2] a ⊗ z
i
, 1 ) ⊗ z
i − 1
⊠ z
i +1
∼
= Hom
A
( z
i
, 1 ) ⊗ z
i − 1
⊠ z
i +1
.
[ κ
i
+ κ
i +1
]
a,
(C.8)
and similarly for other combinations of orientations. In these expressions we display only the
relevant part of the object, and we use the balancings of the object z
ǫ
1
1
⊠ · · · ⊠ z
ǫ
n
n
in steps two