[<< wikibooks] Homological Algebra/Abelian Category
== Exercises ==
Given

f
:
a
→
b

{\displaystyle f:a\to b}
in an

A
b

{\displaystyle Ab}
-enriched category with zero object. Prove that

f
=

0

a
,
b

{\displaystyle f=0_{a,b}}
iff

f

{\displaystyle f}
factors through

0

{\displaystyle \mathbf {0} }
.Given a biproduct

(
a
⊕
b
,

i

1

,

p

1

,

i

2

,

p

2

)

{\displaystyle (a\oplus b,i_{1},p_{1},i_{2},p_{2})}
of

a

{\displaystyle a}
and

b

{\displaystyle b}
. Prove that

(
a
⊕
b
,

i

1

,

i

2

)

{\displaystyle (a\oplus b,i_{1},i_{2})}
is a coproduct of

a

{\displaystyle a}
and

b

{\displaystyle b}
and

(
a
⊕
b
,

p

1

,

p

2

)

{\displaystyle (a\oplus b,p_{1},p_{2})}
is a product of

a

{\displaystyle a}
and

b

{\displaystyle b}
.In an

A
b

{\displaystyle Ab}
-enriched category with zero object, a kernel of

f
:
a
→
b

{\displaystyle f:a\to b}
can be equivalently be characterized as a pullback of

0

→
b

{\displaystyle \mathbf {0} \to b}
along

f

{\displaystyle f}
.