In
homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topol ...
in
mathematics, the homotopy category ''K(A)'' of chain complexes in an
additive category
In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts.
Definition
A category C is preadditive if all its hom-sets are abelian groups and composition of morp ...
''A'' is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of
chain complexes
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...
''Kom(A)'' of ''A'' and the
derived category ''D(A)'' of ''A'' when ''A'' is
abelian; unlike the former it is a
triangulated category In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy categ ...
, and unlike the latter its formation does not require that ''A'' is abelian. Philosophically, while ''D(A)'' turns into isomorphisms any maps of complexes that are
quasi-isomorphisms in ''Kom(A)'', ''K(A)'' does so only for those that are quasi-isomorphisms for a "good reason", namely actually having an inverse up to homotopy equivalence. Thus, ''K(A)'' is more understandable than ''D(A)''.
Definitions
Let ''A'' be an
additive category
In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts.
Definition
A category C is preadditive if all its hom-sets are abelian groups and composition of morp ...
. The homotopy category ''K(A)'' is based on the following definition: if we have complexes ''A'', ''B'' and maps ''f'', ''g'' from ''A'' to ''B'', a chain homotopy from ''f'' to ''g'' is a collection of maps
(''not'' a map of complexes) such that
:
or simply
This can be depicted as:
:
We also say that ''f'' and ''g'' are chain homotopic, or that
is null-homotopic or homotopic to 0. It is clear from the definition that the maps of complexes which are null-homotopic form a group under addition.
The homotopy category of chain complexes ''K(A)'' is then defined as follows: its objects are the same as the objects of ''Kom(A)'', namely
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...
es. Its morphisms are "maps of complexes modulo homotopy": that is, we define an equivalence relation
:
if ''f'' is homotopic to ''g''
and define
:
to be the
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
by this relation. It is clear that this results in an additive category if one notes that this is the same as taking the quotient by the subgroup of null-homotopic maps.
The following variants of the definition are also widely used: if one takes only ''bounded-below'' (''A
n=0 for n<<0''), ''bounded-above'' (''A
n=0 for n>>0''), or ''bounded'' (''A
n=0 for , n, >>0'') complexes instead of unbounded ones, one speaks of the ''bounded-below homotopy category'' etc. They are denoted by ''K
+(A)'', ''K
−(A)'' and ''K
b(A)'', respectively.
A morphism
which is an isomorphism in ''K(A)'' is called a homotopy equivalence. In detail, this means there is another map
, such that the two compositions are homotopic to the identities:
and
.
The name "homotopy" comes from the fact that
homotopic
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
maps of
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s induce homotopic (in the above sense) maps of
singular chain
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...
s.
Remarks
Two chain homotopic maps ''f'' and ''g'' induce the same maps on homology because ''(f − g)'' sends
cycles to
boundaries, which are zero in homology. In particular a homotopy equivalence is a
quasi-isomorphism. (The converse is false in general.) This shows that there is a canonical functor
to the
derived category (if ''A'' is
abelian).
The triangulated structure
The ''shift'' ''A
' of a complex ''A'' is the following complex
:
(note that
),
where the differential is
.
For the cone of a morphism ''f'' we take the
mapping cone. There are natural maps
: