which has as parallel morphism the composition with coev
l
a
at
x
n
.
Again, these morphisms
factorize over the end, and by composing the defining equation (4.43) with the balancing of
x
we see that the block space is also the equalizer
T
fine
,
(
v
)
(
...
x
m
⊠
x
n
⊠
y
)
−→
T
pre
(
...
x
m
⊠
x
n
⊠
y
)
producttext
tildewidest
hol
x
−
−
−
−
−
−
−
−
−
−
−
−
⇒
producttext
(coev
l
)
∗
productdisplay
x
integraldisplay
a
∈A
T
pre
(
...,
x
m
⊠
(
∨
a
⊗
a
)
.x
n
⊠
y
)
.
(4.45)
We use the latter description of the block space to show
Lemma 4.19.
The block functor depends on the choice of a starting point per disk only up to
canonical coherent natural isomorphism.
Proof.
We define canonical natural isomorphisms
Γ
v
′
,v
:
T
fine
,
(
v
)
(
...,
x
m
⊠
x
n
⊠
y
)
−→
T
fine
,
(
v
′
)
(
...,
x
m
⊠
x
n
⊠
y
)
,
(4.46)
for each pair of starting points
v,v
′
per disk, that satisfy the coherence relation Γ
v
′′
,v
′
◦
Γ
v
′
,v
= Γ
v
′′
,v
.
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- Fall '17
