In
homotopy theory, a branch of
mathematics, a Quillen adjunction between two
closed model categories C and D is a special kind of
adjunction between
categories
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)
* ...
that induces an adjunction between the
homotopy categories Ho(C) and Ho(D) via the
total derived functor construction. Quillen adjunctions are named in honor of the mathematician
Daniel Quillen
Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 197 ...
.
Formal definition
Given two closed model categories C and D, a Quillen adjunction is a pair
:(''F'', ''G''): C
D
of
adjoint functor
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
s with ''F'' left adjoint to ''G'' such that ''F'' preserves
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 and trivial cofibrations or, equivalently by the closed model axioms, such that ''G'' preserves
fibrations and trivial fibrations. In such an adjunction ''F'' is called the left Quillen functor and ''G'' is called the right Quillen functor.
Properties
It is a consequence of the axioms that a left (right) Quillen functor preserves
weak equivalences between cofibrant (fibrant) objects. The
total derived functor theorem of Quillen says that the total left derived functor
:L''F'': Ho(C) → Ho(D)
is a left adjoint to the total right derived functor
:R''G'': Ho(D) → Ho(C).
This adjunction (L''F'', R''G'') is called the derived adjunction.
If (''F'', ''G'') is a Quillen adjunction as above such that
:''F''(''c'') → ''d''
with ''c'' cofibrant and ''d'' fibrant is a weak equivalence in D if and only if
:''c'' → ''G''(''d'')
is a weak equivalence in C then it is called a Quillen equivalence of the closed model categories C and D. In this case the derived adjunction is an adjoint
equivalence of categories
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences f ...
so that
:L''F''(''c'') → ''d''
is an isomorphism in Ho(D) if and only if
:''c'' → R''G''(''d'')
is an isomorphism in Ho(C).
References
* {{Cite book , last1=Goerss , first1=Paul G. , author-link1=:de:Paul Goerss , last2=Jardine , first2=John F. , author-link2=Rick Jardine , title=Simplicial Homotopy Theory , publisher=Birkhäuser , location=Basel, Boston, Berlin , series=Progress in Mathematics , isbn=978-3-7643-6064-1 , year=1999 , volume=174 , postscript=
* Philip S. Hirschhorn, Model Categories and Their Localizations, American Mathematical Soc., Aug 24, 2009 - Mathematics - 457 pages
Homotopy theory
Theory of continuous functions
Adjoint functors