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Categories, abelian categories and cohomologies.
Yoshifumi Tsuchimoto
Our treatment here is a (rather strange) mixture of [2],[1]
DEFINITION 03.1
Let
be two functors from a category
to a category
.
A morphism of functros from
to
is a family of morphisms in
:
one for each
, satisfying the following condition: for any morphism
in
, the diagram
is commutative.
DEFINITION 03.2
Let
be a category,
be objects of
.
Then an morphism
is an
isomorphism in
if there exists
such that the relations
hold.
Objects
in a category
are said to be
isomorphic
if there exists at least one isomorphism between them.
Note that by combining the above two definitions, we obtain a definition
of a notion of isomorphisms of functors.
DEFINITION 03.3
A functior
is said to be an
equivalene of category
if there exists a functor
such that the
functor
is isomorphic to
, and the functor
is
isomorphic to
.
If such a thing exists, we say that the two categories are
equivalent.
DEFINITION 03.5
An category
is an
additive category
if it satisfies the following axioms:
 A1.
 Any set
is an additive group.
The composition of morphisms is biadditive.
 A2.
 There exists a null object
.
 A3.
 For any objects
, there exists a biproduct
of
. Namely, there exists a diagram
in
such that
holds.
DEFINITION 03.6
Let
be a category,
, adn
.
An
equalizer
of
is
an arrow
in
which satisfies the
following properties:

.

is ``universal'' amoung morphisms which satisfies (1).
In other words, if
is a morphism in
such that
, then there exists a unique arrow
in
which satisfy
By reversing the directions of arrows above, one may define the notion of
coequalizers
DEFINITION 03.7
Let
be an additive category. Then the equalizer
(respectively, coequalizer) of an arrow
and
is called the
kernel
(respectively,
cokernel) of
.
DEFINITION 03.8
An additive category
is said to be
abelian if it satisfies the
following axioms.
 A41.
 Every morphism
in
has a kernel
.
 A42.
 Every morphism
in
has a
cokernel
.
 A43.
 For any given morphism
, we have
a suitably defined isomorphism
in
.
More precisely,
is a morphism which is defined by the following relations:
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20090515