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In-Depth Information
From the commutativity of the triangle above,
id
A
=
h
◦
η
A
, it must hold, from
Definition
23
, that
h
◦
η
A
id
A
id
A
=
and hence for each ptp arrow
id
A
j
:
A
j
→
A
J
in
,
h
we must have the ptp arrow
h
jj
:
TA
j
→
A
J
in
. Consequently,
id
A
j
=
h
jj
◦
η
A
j
,
that is,
id
A
j
=
TA
j
=
h
jj
◦
η
A
j
=
h
jj
∩
η
A
j
=
h
jj
∩
TA
j
=
h
jj
(from the fact that,
from Proposition
7
,
h
jj
⊆
T(TA
j
)
∩
TA
j
=
TA
j
). Thus, from Proposition
8
,
h
jj
:
h
TA
j
→
. However,
from the fact that the number if ptp arrows in
h
cannot be greater than the number
of ptp arrows in
id
A
(otherwise, the composed arrow
h
A
J
is an isomorphism and
⊇{
h
jj
:
TA
j
→
A
j
|
1
≤
j
≤
m
}
is
A
would have
more ptp arrows than
is
A
, i.e., more ptp arrows than in
id
A
, and we would have that
id
A
◦
η
A
=
h
◦
h
◦
η
A
is
−
A
:
=
, i.e.,
id
A
=
h
◦
η
A
), we obtain this unique monadic T-algebra
h
=
TA
→
A
.
3.2.2 Duality
The following duality theorem shows that for any commutative diagram in
DB
there
is an equal commutative diagram, composed of equal objects and inverted equiva-
lent arrows. This duality property of
DB
is a consequence of the fact that a composi-
tion of simple arrows (i.e., morphisms) is semantically based on the set-intersection
commutativity of information fluxes of these arrows, and the information flux of
inverted arrow is equal to the information flux of the original arrow.
Thus,
any limit diagram
in
DB
has also its equivalent
inverted colimit diagram
with equal objects and, more generally,
any universal property
also has its
equiv-
alent couniversal property
in
DB
. From the fact that all arrows are composed of
simple atomic arrows, it is enough to consider only the atomic arrows. Duality prop-
erties for different cases of complex arrows are presented with details also in point
3 of Corollary
12
in Sect.
3.2.4
.
(
S
0
,
S
1
)
=
:
−→
Theorem 3
There exists a contravariant functor
S
DB
DB
such
that
:
1.
S
0
is an identity function on objects
.
2.
Fo r a n y s i m p l e a r row f
:
A
−→
B with (Tf )
OP
id
R
|
R
∈
f
}:
TB
→
TA
,
{
we define
S
1
(f )
f
OP
A
,
where f
OP
is
−
A
◦
(Tf )
OP
:
B
→
=
◦
is
B
is an in-
verted morphism equivalent to f
(
i
.
e
.,
with f
OP
=
f
).
The inverse of an inverted morphism f
OP
is equal to the original morphism
,
that is
,
(f
OP
)
OP
f
.
For a complex arrow
,
we have
f
OP
f
}
f
OP
=
={
ji
|
f
ji
∈
.
3.
The category
DB
is isomorphic to its dual category
DB
OP
.
is
B
=
is
−
A
∩
=
f
and
f
OP
is
−
A
◦
∩
is
B
=
(Tf )
OP
(Tf )
OP
(Tf )
OP
Proof
=
◦
TA
∩
∩
TB
=
TA
∩
f
∩
TB
=
f
.
(Tf )
OP
