Database Reference
In-Depth Information
For every object
X
in
C
,
F
1
(id
X
)
•
=
id
F(X)
;
For all morphisms
f
:
X
→
Y
and
g
:
Y
→
Z
,
F
1
(g
◦
f)
=
F
1
(g)
◦
F
1
(f )
.
A contravariant functor
F
•
D
is like a covariant functor, except that it “turns
morphisms around” (“reverses all the arrows”). More specifically, every morphism
f
:
C
→
F
0
(X)
in
D
.In
other words, a contravariant functor is a covariant functor from the opposite category
C
op
Y
in
C
must be assigned to a morphism
F
1
(f )
F
0
(Y)
:
X
→
:
→
to
D
. Opposite category
C
op
has the same objects as the category
C
.Given
g
in
C
, its opposite arrow in
C
op
g)
OP
a composition of arrows
f
◦
is
(f
◦
=
g
OP
◦
f
OP
.
Every functor preserves epi-, mono- and isomorphisms.
A locally small category
C
, for any object
X
, has a
representable functor F
X
=
C
(X,
_
)
:
C
→
Set
, such that for any object
Y
,
F
X
(Y)
=
C
(X,Y)
, and any arrow
Y
gives by composition a function
F
X
(f )
C
(X,Y
)
.
f
:
Y
→
:
C
(X,Y)
→
A functor
F
:
C
→
D
is called
full
if for every two objects
A
and
B
of
C
,
F
:
→
C
(A,B)
D
(F(A),F(B))
is a surjection. A functor
F
is called
faithful
if this
map is always injective. A functor
F
is called
endofunctor
if
D
C
.
A functor
F reflects
a property
P
if whenever the
F
-image of something (object,
arrow, etc.) has
P
, then that something has
P
. For example, the faithful functor
reflects epimorphisms and monomorphisms.
It is well known that each endofunctor
F
=
:
C
→
C
defines the algebras and coal-
gebras (the left and right commutative diagrams)
so that the morphism
f
(B,k)
, which represents the commutative dia-
gram on the left, is a morphism between F-algebras
(A,h)
and
(B,k)
. For example,
let
F
:
:
(A,h)
→
Set
→
Set
be the polynomial endofunctor such that for the set
A
of all reals,
A
2
A
2
(where
F(A)
=
A
+
A
+
+
+
is a disjoint union) we can represent the algebra
A
2
A
2
for operators
{
exp
:
A
→
A,
log
:
A
→
A,
max
:
→
A,
min
:
→
A
}
, where
A
2
is the Cartesian product
A
×
A
, by the morphism
h
=[
exp
,
log
,
max
,
min
]:
A
2
A
2
)
(A
A
in
Set
, that is, by the F-algebra
(A,h)
.
Analogously, the morphism
f
1
:
+
A
+
+
→
(B,k
1
)
, which represents the com-
mutative diagram on the right, is a morphism between F-coalgebras
(A,h
1
)
and
(B,k
1
)
.
Let
G
(A,h
1
)
→
C
be a functor, and
C
an object of
C
.A
universal arrow from C to
G
is a pair
(D,g)
where
D
is an object of
D
and
g
:
D
→
G(D)
an morphism in
C
such that, for any object
D
of
D
and morphism
f
:
C
→
G(D
)
, there exists a
unique morphism
f
:
C
→
G(f)
D
in
D
such that
f
:
D
→
=
◦
g
. Diagrammatically,
