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Chapter 10
Introducing Homological
Algebra
Roughly speaking, homological algebra consists of (A) that part of algebra that is funda
mental in building the foundations of algebraic topology, and (B) areas that arise naturally
in studying (A).
10.1 Categories
We have now encountered many algebraic structures and maps between these structures.
There are ideas that seem to occur regardless of the particular structure under consider
ation. Category theory focuses on principles that are common to all algebraic systems.
10.1.1 Defnitions and Comments
A
category
C
consists of
objects
A,B,C,.
..
and
morphisms
f
:
A
→
B
(where
A
and
B
are objects). If
f
:
A
→
B
and
g
:
B
→
C
are morphisms, we have a notion of
composition
,
in other words, there is a morphism
gf
=
g
◦
f
:
A
→
C
, such that the following axioms
are satisFed.
(i)
Associativity
:I
f
f
:
A
→
B
,
g
:
B
→
C
,
h
:
C
→
D
, then (
hg
)
f
=
h
(
);
(ii)
Identity
: ±or each object
A
there is a morphism 1
A
:
A
→
A
such that for each
morphism
f
:
A
→
B
, we have
f
1
A
=1
B
f
=
f
.
A remark for those familiar with set theory: ±or each pair (
A,B
) of objects, the
collection of morphisms
f
:
A
→
B
is required to be a set rather than a proper class.
We have seen many examples:
1.
Sets
: The objects are sets and the morphisms are functions.
2.
Groups
: The objects are groups and the morphisms are group homomorphisms.
3.
Rings
: The objects are rings and the morphisms are ring homomorphisms.
1
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CHAPTER 10. INTRODUCING HOMOLOGICAL ALGEBRA
4.
Fields
: The objects are felds and the morphisms are feld homomorphisms [= feld
monomorphisms; see (3.1.2)].
5.
Rmod
: The objects are leFt
R
modules and the morphisms are
R
module homomor
phisms. IF we use right
R
modules, the corresponding category is called
modR
.
6.
Top
: The objects are topological spaces and the morphisms are continuous maps.
7.
Ab
: The objects are abelian groups and the the morphisms are homomorphisms From
one abelian group to another.
A morphism
f
:
A
→
B
is said to be an
isomorphism
iF there is an inverse morphism
g
:
B
→
A
, that is,
gf
=1
A
and
fg
B
.I
n
Sets
, isomorphisms are bijections, and
in
, isomorphisms are homeomorphisms. ±or the other examples, an isomorphism is
a bijective homomorphism, as usual.
In the category oF sets, a Function
f
is injective i²
f
(
x
1
)=
f
(
x
2
) implies
x
1
=
x
2
. But
in an abstract category, we don’t have any elements to work with; a morphism
f
:
A
→
B
can be regarded as simply an arrow From
A
to
B
. How do we generalize injectivity to an
arbitrary category? We must give a defnition that does not depend on elements oF a set.
Now in
Sets
,
f
is injective i² it has a leFt inverse; equivalently,
f
is
left cancellable
, i.e.
iF
fh
1
=
2
, then
h
1
=
h
2
. This is exactly what we need, and a similar idea works For
surjectivity, since
f
is surjective i²
f
is
right cancellable
, i.e.,
h
1
f
=
h
2
f
implies
h
1
=
h
2
.
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 Spring '11
 sdd
 Logic, Algebra, Algebraic Topology

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