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 )...
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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...
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Algebraic K-theory (redirect from Quillen's +-construction)
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...
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_{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...
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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...
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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...
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Kan complexes and CW complexes being given by the geometric realization and the singular functor (Milnor's theorem). The Kan–Quillen model structure...
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Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem or the full embedding theorem, is a result about abelian categories; it essentially...
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the index theorem, especially in formulating related fixed-point theorems, in particular the so-called 'Woods Hole fixed-point theorem', a combination...
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spaces, and are the successive quotients of the terms of the Bott periodicity clock. These equivalences immediately yield the Bott periodicity theorems. The...
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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...
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Projective module (redirect from Rank of a projective module)
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...
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concerns a category with cofibrations and weak equivalences; such a category is called a Waldhausen category and generalizes Quillen's exact category. A cofibration...
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Quotient of an abelian category (category Pages that use a deprecated format of the math tags)
R} , and B {\displaystyle {\cal {B}}} is some localizing subcategory of Mod ( R ) {\displaystyle \operatorname {Mod} (R)} . Daniel Quillen's algebraic...
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Q-construction (redirect from Quillen's Q-construction)
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...
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new proofs and generalizations of the Atiyah-Singer index theorem. Among these generalizations are index theorems based on spectral triples and deformation...
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Exact functor (section Properties and theorems)
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...
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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...
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K-theory (section K0 of a field)
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 )...
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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...
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(1969), Theorem 2.3.5; Sjödin (1980), Theorem 1. Quillen (1970), section 7. Avramov, Luchezar; Halperin, Stephen (1986), "Through the looking glass: a dictionary...
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Algebraic topology (section Important theorems)
(Gives a broad view of higher-dimensional van Kampen theorems involving multiple groupoids). Brown, R.; Razak, A. (1984), "A van Kampen theorem for unions...
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Simplicial set (redirect from Geometric realization of a simplicial set)
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...
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3-category (section Strict and weak 3-categories)
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...
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as of September 2022[update]. The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic...
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_{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 )...
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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...
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Clark Barwick (category University of Pennsylvania School of Arts and Sciences alumni)
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...
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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...
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Glossary of algebraic topology (section B)
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...
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