Database Reference
In-Depth Information
It is easy to verify also that in extended symmetric categories the following is valid:
=
T
e
τ
I
F
st
;
ψ
•
ϕ
−
1
F
st
,
=
ϕ
−
1
S
nd
•
T
e
ψ
τ
I
S
nd
,
τ
−
1
τ
;
where
τ
I
:
I
C
−→
I
C
is an identity natural transformation (for any object
A
in
C
,
τ
I
(A)
=
id
A
).
Example 1
The
Set
is an extended symmetric category: given any function
f
:
A
−→
B
, the conceptualized object of this function is the graph of this function
(which is a set),
f
=
B
T
(f )
{
(x,f (x))
|
x
∈
A
}
.
is defined as an associative composition of binary
relations (graphs of functions),
B
T
(g
◦
f)
={
(x,(g
◦
f )(x))
|
x
∈
A
}={
(y,g(y))
|
y
∈
B
}◦{
(x,f (x))
|
x
∈
A
}=
B
T
(g)
∗
B
T
(f )
.
Set
is also conceptually closed by the functor
T
e
such that for any object
J(f)
=
A,B,f
The composition of objects
∗
,
T
e
(J(f ))
B
T
(f )
={
(x,f (x))
|
x
∈
A
}
, and for any arrow
(k
1
;
k
2
)
:
J(f)
→
J(g)
, the component
T
e
is defined by:
for any
x,f(x)
∈
T
e
J(f)
,T
e
(k
1
;
k
2
)
x,f(x)
=
k
1
(x),k
2
f(x)
.
It is easy to verify the compositional property for
T
e
, and that
T
e
(id
A
;
=
id
T
e
(J(f ))
. For example,
Set
is also an extended symmetric category such that for
any object
J(f)
id
B
)
=
A,B,f
in
Set
↓
Set
,
τ(J(f))
:
A
B
T
(f )
is an epimorphism
(x,f (x))
, while
τ
−
1
(J(f ))
such that for any
x
∈
A
,
τ(J(f ))(x)
=
:
B
T
(f )
→
B
is
B
T
(f ),τ
−
1
(J(f ))(x,f (x))
the second projection, such that for any
(x,f (x))
=
f(x)
. Thus, each arrow in
Set
is a composition of an epimorphism (surjective func-
tion) and a monomorphism (injective function).
∈
The categorial symmetry can be better understood by considering a category
C
as a two-sorted algebra
Alg
C
=
((Ob
C
, Mor
C
),Σ
C
)
(two-sorted carrier set is com-
posed of its objects and its morphisms) with the signature
Σ
C
={
dom, cod, id,
◦}∪
{
o
i
|
0
≤
i
≤
n
}
, where
1.
dom, cod
:
Mor
C
→
Ob
C
are two operations such that for any morphism
f
:
A
→
B
,
dom(f )
=
A,cod(f )
=
B
;
2.
id
:
Ob
C
→
Mor
C
is the operations such that for any object
A
,
id(A)
=
id
A
:
A
→
A
is the identity arrow for this object;
Mor
C
→
3.
◦:
Mor
C
is a
partial
function, such that for any two morphisms
f,g
,
f
◦
g
is defined if
dom(f )
=
cod(g)
, with the following set of equations:
•
(associativity)
(f
◦
g)
◦
h
=
f
◦
(g
◦
h)
,if
dom(g)
=
cod(h)
and
dom(f )
=
cod(g)
,
id(dom(f ))
=
f
and
id(cod(f ))
◦
f
=
f
;
4. For each operator
o
i
∈
•
(identity)
f
◦
Ob
ar(o
i
)
C
Ob
C
(as,
for example, the product, coproduct, etc.), or the composition of morphisms
o
i
:
Mor
ar(o
i
)
C
Σ
C
, the composition of objects
o
i
:
→
Mor
C
(as, for example, the product, coproduct, the various pairings
(structural operations)
(,),
→
,etc.).
Consequently, the categorial symmetry can be done by the following corollary:
,
,
[
,
]
