homotopy theory In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipli ...

, a branch of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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: General uses *Classification, the general act of allocating things to classes/categories Philosophy *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) *Category (Vais ...

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 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 1978. Fr ...

Formal definition

Given two closed model categories C and D, a Quillen adjunction is a pair :(''F'', ''G''): C

\leftrightarrows

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 k ...

s with ''F'' left adjoint to ''G'' such that ''F'' preserves

cofibration In mathematics, in particular homotopy theory, a continuous mapping between topological spaces :i: A \to X, is a ''cofibration'' if it has the homotopy extension property with respect to all topological spaces S. That is, i is a cofibration if f ...

s and trivial cofibrations or, equivalently by the closed model axioms, such that ''G'' preserves

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 ma ...

s 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 Category (mathematics), categories that establishes that these categories are "essentially the same". There are numerous examples of cate ...

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

* 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