Quillen adjunction

In homotopy theory, a branch of mathematics, a Quillen adjunction between two C and D is a special kind of adjunction between that induces an adjunction between the Ho(C) and Ho(D) via the total derived functor construction. Quillen adjunctions are named in honor of the mathematician Daniel Quillen. Given two closed model categories C and D, a Quillen adjunction is a pair (F, G): C D of adjoint functors with F left adjoint to G such that F preserves cofibrations 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. 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 LF: Ho(C) → Ho(D) is a left adjoint to the total right derived functor RG: Ho(D) → Ho(C). This adjunction (LF, RG) 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 so that LF(c) → d is an isomorphism in Ho(D) if and only if c → RG(d) is an isomorphism in Ho(C).

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (1)

Related concepts (2)

Related courses (1)

Related lectures (32)

Quillen adjunction

Model category

In mathematics, particularly in homotopy theory, a model category is a with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes ( theory). The concept was introduced by . In recent decades, the language of model categories has been used in some parts of algebraic K-theory and algebraic geometry, where homotopy-theoretic approaches led to deep results.

Serre model structure on Top MATH-436: Homotopical algebra

Explores the Serre model structure on Top, focusing on right and left homotopy.

Transfer of Model Structures MATH-436: Homotopical algebra

Covers the transfer of model structures through adjunctions in the context of model categories.

Homotopy Theory of Chain Complexes MATH-436: Homotopical algebra

Explores the model structure on chain complexes over a field.

Catégories simpliciales enrichies et K-théorie de Waldhausen

Ilias Amrani

This thesis, which presents a new approach to the algebraic K-theory, is divided into two parts. The first one is devoted to the category of small simplicial categories. First, we construct a new mode

EPFL2010

This course will provide an introduction to model category theory, which is an abstract framework for generalizing homotopy theory beyond topological spaces and continuous maps. We will study numerous