Database Reference
In-Depth Information
T
is a V-functor: for each pair of objects
A
and
B
there exists an identity map
(see above)
f
AB
:
(T B)
TA
, subject to the compatibility with composition
m
and with the identities expressed by the commutativity of the diagram
B
A
−→
that is,
f
AC
◦
m
A,B,C
=
m
TA,TB,TC
◦
(f
BC
⊗
f
AB
)
and
j
TA
=
f
AB
◦
j
A
.Fromthe
fact that
f
BC
and
f
AB
are the identity arrows,
(f
BC
⊗
f
AB
)
is identity as well, so it
=
1
≤
j
≤
m
,m
is enough to show that
m
TA,TB,TC
=
m
A,B,C
. In fact, for any
A
≥
1,
=
1
≤
i
≤
k
,k
=
1
≤
l
≤
n
,n
1, and from
C
B
C
,
B
A
B
≥
1 and
C
≥
B
⊗
A
⊗
B
,
we obtain
m
A,B,C
=
in
ilji
1
:
(B
i
⊗
C
l
)
⊗
(A
j
⊗
B
i
1
)
→
A
j
⊗
C
l
|
≤
l
≤
n
=
in
ilji
1
:
(T B
i
∩
TC
l
)
∩
(T A
j
∩
TB
i
1
)
→
TA
j
∩
TC
l
|
1
≤
j
≤
m,
1
≤
i,i
1
≤
k,
1
≤
l
≤
n
=
in
ilji
1
:
T(TB
i
)
∩
T(TC
l
)
∩
T(TA
j
)
∩
T(TB
i
1
)
→
T(TA
j
)
∩
T(TC
l
)
|
1
≤
j
≤
m,
1
≤
i,i
1
≤
k,
1
n
1
≤
j
≤
m,
1
≤
i,i
1
≤
k,
1
≤
l
≤
=
in
ilji
1
:
(T B
i
⊗
TC
l
)
⊗
(T A
j
⊗
TB
i
1
)
→
TA
j
⊗
TC
l
n
|
1
≤
j
≤
m,
1
≤
i,i
1
≤
k,
1
≤
l
≤
m
TA,TB,TC
=
,
that is,
m
TA,TB,TC
=
m
A,B,C
.
It is easy to verify that also natural transformations
T
2
η
:
I
DB
−→
T
and
μ
:
−→
T
satisfy the V-naturality condition (V-natural transformation
η
and
μ
are an
Ob
DB
-
indexed family of components
δ
A
:
Υ
TA
in
DB
(for
η
,
δ
A
:
Υ
(T A)
A
,
(T A)
A
TA
T
2
A
,
(T A)
T
2
A
TA
). This map
f
AB
is equal also for the endofunctor identity
I
DB
, and for the endofunctor
T
2
, because
B
A
≈
TA
; while for
μ
,
δ
A
:
Υ
≈
(T
2
B)
T
2
A
.
(T B)
TA
⊗
=
A
B
