Database Reference
In-Depth Information
•
Category
Set
where the objects are sets and the arrows are functions from one
set to another.
•
A preorder is a set
X
together with a binary relation
≤
which is reflexive (i.e.,
≤
∈
≤
≤
≤
x
x
for all
x
X
), and transitive (i.e.,
x
y
and
y
z
imply
x
z
for all
∈
x,y,z
X
). This can be seen as a category, with set of objects
X
and for every
pair of objects
(x,y)
such that
x
≤
y
, exactly one arrow
x
→
y
.
•
Any set
X
can be seen as a discrete category, with a set of objects
X
and only
with the identity morphisms.
Properties of morphisms:
Morphisms can have any of the following properties. A morphism
f
:
A
→
B
is a
•
Monomorphism
(or monic) if
f
◦
g
1
=
f
◦
g
2
implies
g
1
=
g
2
for all morphisms
g
1
,g
2
:
X
→
A
. It is denoted by
f
:
A
→
B
.
•
Epimorphism
(or epic) if
g
1
◦
f
=
g
2
◦
f
implies
g
1
=
g
2
for all morphisms
g
1
,g
2
:
B
→
X
. It is denoted by
f
:
A
B
. This epic is called
split
if there is
g
:
B
→
A
such that
f
◦
g
=
id
B
(then
f
is a
retraction
of
g
).
•
Isomorphism
if there exists a morphism
g
:
B
→
A
such that
f
◦
g
=
id
B
and
B
the
isomorphism of two objects. If
f
is split epic and monic,
f
is isomorphism.
For example, in
Set
the epimorphisms are surjective functions, the monomorphisms
are injective functions, the isomorphisms are bijective functions.
The definition of epic is
dual
to the definition of monic. With
C
OP
we denote
the opposite category of
C
, where all arrows are reversed. Thus,
f
is monic in the
category
C
iff
f
OP
is epic in
C
OP
, and vice versa. In general, given a property
P
of an object, arrow, diagram, etc., we can associate with
P
the dual property
P
OP
:
The object or arrow has a property
P
in
C
iff it has
P
OP
in
C
OP
.
For example, if
g
g
◦
f
=
id
A
. It is denoted by
f
:
A
B
. Often we denote simply by
A
f
is monic then
f
is monic. From this, by duality, if
f
OP
◦
◦
g
OP
is epic then
f
OP
is epic.
An object
X
is called
terminal
if for any object
Y
there is exactly one morphism
from it into
X
in the category. Dually, one object
X
is called
initial
if for any object
Y
there is exactly one morphism from
X
to
Y
in the category. An object
X
is called
a
zero
object if it is both terminal and initial. For example, in
Set
the empty set is
initial, while any singleton set is (up to isomorphism) a terminal object.
Given two categories
C
and
D
, we can define the
product category
C
×
D
which
(X
,Y
)
pairs
has as objects pairs
(X,Y)
∈
Ob
C
×
Ob
D
, and as arrows
(X,Y)
→
(f,g)
with
f
:
X
→
X
in
C
, and
g
:
Y
→
Y
in
D
.
Functors
are structure-preserving maps between categories. They can be thought
of as morphisms in the category
Cat
of all (small) categories as objects, and all
functors as morphisms
A (covariant) functor
F
from a category
C
to a category
D
, written
F
:
C
→
D
,
consists of a pair of functions
F
=
(F
0
,F
1
)
,
F
0
Ob
D
and
F
1
:
Ob
C
→
:
Mor
C
→
Mor
D
(in what follows, we will simply use
F
for both functions as well):
•
For each object
X
in
C
, an object
F
0
(X)
in
D
; and
Y
in
C
, a morphism
F
1
(f )
F
0
(X)
F
0
(Y)
,
•
For each morphism
f
:
X
→
:
→
such that the following two properties hold:
