In
mathematics, a weak equivalence is a notion from
homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
atic definition of a
model category.
A model category is a
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
*C ...
with classes of
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s called weak equivalences,
fibrations, and
cofibration In mathematics, in particular homotopy theory, a continuous mapping
:i: A \to X,
where A and X are topological spaces, is a cofibration if it lets homotopy classes of maps ,S/math> be extended to homotopy classes of maps ,S/math> whenever a map ...
s, satisfying several axioms. The associated
homotopy category In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed b ...
of a model category has the same objects, but the morphisms are changed in order to make the weak equivalences into
isomorphisms. It is a useful observation that the associated homotopy category depends only on the weak equivalences, not on the fibrations and cofibrations.
Topological spaces
Model categories were defined by
Quillen as an axiomatization of homotopy theory that applies to
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, but also to many other categories in
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
and
geometry. The example that started the subject is the category of topological spaces with
Serre fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in postnikov-systems or obstruction theory.
In this article, all map ...
s as fibrations and weak homotopy equivalences as weak equivalences (the cofibrations for this model structure can be described as the
retracts of relative cell complexes ''X'' ⊆ ''Y''). By definition, a
continuous mapping
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
''f'': ''X'' → ''Y'' of spaces is called a weak homotopy equivalence if the induced function on sets of
path components
:
is
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
, and for every point ''x'' in ''X'' and every ''n'' ≥ 1, the induced
homomorphism
:
on
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homot ...
s is bijective. (For ''X'' and ''Y''
path-connected, the first condition is automatic, and it suffices to state the second condition for a single point ''x'' in ''X''.)
For
simply connected topological spaces ''X'' and ''Y'', a map ''f'': ''X'' → ''Y'' is a weak homotopy equivalence if and only if the induced homomorphism ''f''
*: ''H''
''n''(''X'',Z) → ''H''
''n''(''Y'',Z) on
singular homology groups is bijective for all ''n''. Likewise, for simply connected spaces ''X'' and ''Y'', a map ''f'': ''X'' → ''Y'' is a weak homotopy equivalence if and only if the pullback homomorphism ''f''*: ''H''
''n''(''Y'',Z) → ''H''
''n''(''X'',Z) on
singular cohomology is bijective for all ''n''.
Example: Let ''X'' be the set of natural numbers and let ''Y'' be the set ∪ , both with the
subspace topology from the
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
. Define ''f'': ''X'' → ''Y'' by mapping 0 to 0 and ''n'' to 1/''n'' for positive integers ''n''. Then ''f'' is continuous, and in fact a weak homotopy equivalence, but it is not a
homotopy equivalence
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 defo ...
.
The homotopy category of topological spaces (obtained by inverting the weak homotopy equivalences) greatly simplifies the category of topological spaces. Indeed, this homotopy category is
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
*Equivalence class (music)
*''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*'' Equival ...
to the category of
CW complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
es with morphisms being
homotopy classes of continuous maps.
Many other model structures on the category of topological spaces have also been considered. For example, in the Strøm model structure on topological spaces, the fibrations are the
Hurewicz fibrations and the weak equivalences are the homotopy equivalences.
Chain complexes
Some other important model categories involve
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. Let ''A'' be a
Grothendieck abelian category In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957English translation in order to develop the machinery of homological algebra for modules and for sheaves ...
, for example the category of
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
s over a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
or the category of
sheaves of
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s on a topological space. Define a category ''C''(''A'') with objects the complexes ''X'' of objects in ''A'',
:
and morphisms the
chain map
A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A ...
s. (It is equivalent to consider "cochain complexes" of objects of ''A'', where the numbering is written as
:
simply by defining ''X''
''i'' = ''X''
−''i''.)
The category ''C''(''A'') has a model structure in which the cofibrations are the
monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y.
In the more general setting of category theory, a monomorphism ...
s and the weak equivalences are the
quasi-isomorphism
In homological algebra, a branch of mathematics, a quasi-isomorphism or quism is a morphism ''A'' → ''B'' of chain complexes (respectively, cochain complexes) such that the induced morphisms
:H_n(A_\bullet) \to H_n(B_\bullet)\ (\text H^n(A^\bul ...
s.
[Beke (2000), Proposition 3.13.] By definition, a chain map ''f'': ''X'' → ''Y'' is a quasi-isomorphism if the induced homomorphism
:
on
homology is an isomorphism for all integers ''n''. (Here ''H''
''n''(''X'') is the object of ''A'' defined as the
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learni ...
of ''X''
''n'' → ''X''
''n''−1 modulo the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of ''X''
''n''+1 → ''X''
''n''.) The resulting homotopy category is called the
derived category
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proc ...
''D''(''A'').
Trivial fibrations and trivial cofibrations
In any model category, a fibration that is also a weak equivalence is called a trivial (or acyclic) fibration. A cofibration that is also a weak equivalence is called a trivial (or acyclic) cofibration.
Notes
References
*
*
*
*
{{DEFAULTSORT:Weak Equivalence
Homotopy theory
Homological algebra
Equivalence (mathematics)