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arrows), the composition
of objects can be a (not necessarily) commutative binary
operator as well because the objects have no directionality.
The objects obtained from arrows will be called
conceptualized
objects. For ex-
ample, let us consider the polynomial function
f(x)
∗
(x
2
=
−
3
x
+
1
)
, with the
domain being the closed interval of reals from 0 to 1, and codomain
R
of all reals. It
is an arrow
f
in the category
Set
. The conceptualized object obtained
from this arrow, denoted by
f
,isthe
set
(hence an object in
Set
) equal to the graph
of this polynomial function. That is,
f
:[
0
,
1
]→R
Ob
Set
.
Let us now formally introduce the following definitions [
44
] for the categorial
symmetry:
={
(x,f (x))
|
x
∈[
0
,
1
]}∈
Definition 4
(C
ATEGORIAL SYMMETRY
)L t
C
be a category with a function
B
T
:
A
(the “representability” principle), and with an associative composition operator for
objects
Mor
C
−→
Ob
C
such that for each identity arrow
id
A
:
A
→
A
,
B
T
(id
A
)
∗
such that, for any composition
g
◦
f
∈
Mor
C
of arrows,
B
T
(g)
∗
B
T
(f )
=
B
T
(g
◦
f)
.
The
conceptualized
object
B
T
(f )
of an arrow
f
:
A
−→
B
in
C
will be denoted
by
f
.
Remark
This symmetry property allows us to consider all properties of an arrow
as properties of objects and their compositions. For a given category
C
, its “ar-
row category” is denoted by
C
↓
C
. Objects of this arrow category are the triples
B
is a morphism in
C
, and each morphism in this arrow
category is a couple of two morphisms
(k
1
;
A,B,f
where
f
:
A
→
k
2
)
:
A,B,f
→
C,D,g
such that
k
2
◦
f
=
g
◦
k
1
:
A
→
D
.
Let us introduce, for a category
C
and its arrow category
C
↓
C
, an encapsulation
operator
J
:
Mor
C
−→
Ob
C
↓
C
and its inverse
ψ
, such that for any arrow
f
:
A
−→
B
,
J(f)
f
.
We denote the first and the second comma functorial projections by
F
st
,S
nd
:
=
A,B,f
is its corresponding object in
C
↓
C
and
ψ(
A,B,f
)
=
A
,B
,g
(
C
↓
C
)
−→
C
such that for any arrow
(k
1
;
k
2
)
:
A,B,f
→
in
C
↓
C
k
1
in
C
) we have that
F
st
(
A
,
F
st
(k
1
;
(i.e., when
k
2
◦
k
1
,
S
nd
(
A,B,f
)
=
B
and
S
nd
(k
1
;
k
2
)
=
k
2
. It is easy to extend the operator
ψ
into a
natural transformation
ψ
f
=
g
◦
A,B,f
)
=
k
2
)
=
S
nd
such that its component for an object
J(f)
:
F
st
in
C
↓
C
is the arrow
ψ
J(f)
=
ψ(J(f))
=
f
. We denote the diagonal functor by
:
0
(A)
C
.
An important subset of symmetric categories are the Conceptually Closed and
Extended symmetric categories, as follows:
−→
(
C
↓
C
)
such that, for any object
A
in a category
C
,
=
A,A,id
A
Definition 5
[
44
]
A conceptually closed
category is a symmetric category
C
with a
functor
T
e
=
(T
e
,T
e
)
:
(
C
C
such that
T
e
=
B
T
ψ
, i.e.,
B
T
=
T
e
J
, with
a natural isomorphism
ϕ
:
T
e
◦
I
C
, where
I
C
is an identity functor for
C
.
C
is an
extended symmetric
category if
τ
−
1
↓
C
)
−→
S
nd
, for vertical
•
τ
=
ψ
:
F
st
T
e
and
τ
−
1
S
nd
.
composition of natural transformations
τ
:
F
st
:
T
e
