statements of the theorems are as follows. Quillen's Theorem A—If f : C → D {\displaystyle f:C\to D} is a functor such that the classifying space B ( d ↓ f )...

College, Oxford. Quillen retired at the end of 2006. He died from complications of Alzheimer's disease on April 30, 2011, aged 70. Quillen's best known contribution...

K-theory for a general variety. The first definition of higher K-theory to be widely accepted was Daniel Quillen's. As part of Quillen's work on the Adams...

_{i}(B^{+}{\text{f-gen-Mod}}_{R})} , where B + = Ω B Q {\displaystyle B^{+}=\Omega BQ} is given by Quillen's Q-construction. If R is a regular ring (i.e., has finite...

This is a list of notable theorems. Lists of theorems and similar statements include: List of algebras List of algorithms List of axioms List of conjectures...

as a relation between the sphere spectrum and the classifying spaces of the symmetric groups via Quillen's plus construction. The mapping space Map 0...

Kan complexes and CW complexes being given by the geometric realization and the singular functor (Milnor's theorem). The Kan–Quillen model structure...

Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem or the full embedding theorem, is a result about abelian categories; it essentially...

the index theorem, especially in formulating related fixed-point theorems, in particular the so-called 'Woods Hole fixed-point theorem', a combination...

spaces, and are the successive quotients of the terms of the Bott periodicity clock. These equivalences immediately yield the Bott periodicity theorems. The...

scheme X with action of a linear algebraic group G, via Quillen's Q-construction; thus, by definition, K i G ( X ) = π i ( B + Coh G ⁡ ( X ) ) . {\displaystyle...

rings over a field: this is the Quillen–Suslin theorem. Any free module is projective. The converse is true in the following cases: if R is a field or skew...

concerns a category with cofibrations and weak equivalences; such a category is called a Waldhausen category and generalizes Quillen's exact category. A cofibration...

R} , and B {\displaystyle {\cal {B}}} is some localizing subcategory of Mod ⁡ ( R ) {\displaystyle \operatorname {Mod} (R)} . Daniel Quillen's algebraic...

In algebra, Quillen's Q-construction associates to an exact category (e.g., an abelian category) an algebraic K-theory. More precisely, given an exact...

new proofs and generalizations of the Atiyah-Singer index theorem. Among these generalizations are index theorems based on spectral triples and deformation...

that a functor is pro-representable if and only if it is left exact, under some mild conditions on the category C. The exact functors between Quillen's exact...

group scheme G. It is a theorem of Quillen that G ≅ Z [ b 1 , b 2 , … ] {\displaystyle G\cong \mathbb {Z} [b_{1},b_{2},\dots ]} and assigns to every ring...

with action of a linear algebraic group G {\displaystyle G} , via Quillen's Q-construction; thus, by definition, K i G ( X ) = π i ( B + Coh G ⁡ ( X )...

In mathematics, and especially differential geometry, the Quillen metric is a metric on the determinant line bundle of a family of operators. It was introduced...

(1969), Theorem 2.3.5; Sjödin (1980), Theorem 1. Quillen (1970), section 7. Avramov, Luchezar; Halperin, Stephen (1986), "Through the looking glass: a dictionary...

(Gives a broad view of higher-dimensional van Kampen theorems involving multiple groupoids). Brown, R.; Razak, A. (1984), "A van Kampen theorem for unions...

spaces of categories, and in particular by Quillen's work of algebraic K-theory. In this work, which earned him a Fields Medal, Quillen developed surprisingly...

f:x\to y} in a strict 2-category C and g : a → b {\displaystyle g:a\to b} in D, the usual product is given as f × g : ( x , a ) → ( y , b ) {\displaystyle...

as of September 2022[update]. The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic...

_{p}{\text{CH}}^{p}(X,n)\otimes \mathbb {Q} } giving the relation to Quillen's higher algebraic K-theory. Note that the maps K n M ( F ) → K n ( F )...

seen from Matsumoto's theorem, in higher degrees it was computed by Quillen in conjunction with his work on the Adams conjecture. A different proof was...

Quillen's Theorem B as well as Quillen's dévissage argument in the process. Much of his recent work has concerned equivariant algebraic K-theory and equivariant...

subcomplexes A and B, then the inclusion f: (A,A∩B) → (X,B) induces an isomorphism h i ( A , A ∩ B ) → f ∗ h i ( X , B ) {\displaystyle h_{i}(A,A\cap B){\overset...

the fibers are homotopy equivalent to each other. Quillen 1.  Daniel Quillen 2.  Quillen’s theorem says that π ∗ M U {\displaystyle \pi _{*}MU} is the...