Quillen K-theory
E912805
Quillen K-theory is a sophisticated algebraic K-theory framework defined via higher K-groups of exact or Waldhausen categories, providing deep invariants of rings, schemes, and topological spaces that generalize and extend earlier constructions such as Milnor K-theory.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Quillen K-theory canonical | 4 |
| algebraic K-theory | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T11219367 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Quillen K-theory Context triple: [Milnor K-theory, relatedTo, Quillen K-theory]
-
A.
Milnor K-theory
Milnor K-theory is an algebraic K-theory constructed from fields using tensor powers of their multiplicative groups modulo Steinberg relations, playing a central role in modern algebraic geometry and number theory.
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B.
Introduction to Algebraic K-Theory
Introduction to Algebraic K-Theory is a foundational graduate-level textbook by John Milnor that systematically develops the basic concepts and techniques of algebraic K-theory in a concise and influential style.
-
C.
K-theory
K-theory is a branch of algebraic topology and algebraic geometry that studies vector bundles and generalized cohomology theories using algebraic and categorical methods.
-
D.
Segal conjecture
The Segal conjecture is a fundamental result in algebraic topology that relates the Burnside ring of a finite group to the stable cohomotopy of its classifying space, profoundly influencing equivariant stable homotopy theory.
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E.
“K-Theory” (book with Friedrich Hirzebruch and others)
“K-Theory” is a foundational mathematical monograph co-authored by Michael Atiyah, Friedrich Hirzebruch, and others that systematically develops topological K-theory and its applications in geometry and topology.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Quillen K-theory Target entity description: Quillen K-theory is a sophisticated algebraic K-theory framework defined via higher K-groups of exact or Waldhausen categories, providing deep invariants of rings, schemes, and topological spaces that generalize and extend earlier constructions such as Milnor K-theory.
-
A.
Milnor K-theory
Milnor K-theory is an algebraic K-theory constructed from fields using tensor powers of their multiplicative groups modulo Steinberg relations, playing a central role in modern algebraic geometry and number theory.
-
B.
Introduction to Algebraic K-Theory
Introduction to Algebraic K-Theory is a foundational graduate-level textbook by John Milnor that systematically develops the basic concepts and techniques of algebraic K-theory in a concise and influential style.
-
C.
K-theory
K-theory is a branch of algebraic topology and algebraic geometry that studies vector bundles and generalized cohomology theories using algebraic and categorical methods.
-
D.
Segal conjecture
The Segal conjecture is a fundamental result in algebraic topology that relates the Burnside ring of a finite group to the stable cohomotopy of its classifying space, profoundly influencing equivariant stable homotopy theory.
-
E.
“K-Theory” (book with Friedrich Hirzebruch and others)
“K-Theory” is a foundational mathematical monograph co-authored by Michael Atiyah, Friedrich Hirzebruch, and others that systematically develops topological K-theory and its applications in geometry and topology.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic K-theory
ⓘ
mathematical theory ⓘ |
| appliesTo |
Waldhausen categories
ⓘ
exact categories ⓘ rings ⓘ schemes ⓘ topological spaces ⓘ |
| characteristicProperty |
excision in K-theory
ⓘ
homotopy invariance in suitable settings ⓘ localization sequences ⓘ long exact sequences of K-groups ⓘ |
| defines | higher K-groups ⓘ |
| definesGroup |
K0
ⓘ
K1 ⓘ K2 ⓘ Kn for n≥0 ⓘ |
| extends |
Bass K-theory
ⓘ
Grothendieck group K0 ⓘ
linked to:
Grothendieck group
|
| field |
algebraic geometry
ⓘ
algebraic topology ⓘ category theory ⓘ homological algebra ⓘ |
| framework | homotopy-theoretic ⓘ |
| generalizes | Milnor K-theory ⓘ |
| hasApplication |
Bloch–Kato conjecture
ⓘ
Lichtenbaum–Quillen conjecture ⓘ
linked to:
Quillen–Lichtenbaum conjecture
algebraic cycles ⓘ higher regulators ⓘ motivic cohomology ⓘ number theory ⓘ topological cyclic homology ⓘ |
| influenced |
Waldhausen K-theory
ⓘ
higher category theory ⓘ motivic homotopy theory ⓘ |
| introducedBy | Daniel Quillen ⓘ |
| introducedInYear | 1972 ⓘ |
| namedAfter | Daniel Quillen ⓘ |
| relatedConcept |
Waldhausen K-theory
ⓘ
algebraic K-theory of rings ⓘ algebraic K-theory of schemes ⓘ |
| studiesInvariant |
exact sequences
ⓘ
projective modules ⓘ vector bundles ⓘ |
| usedTo |
define K-theory of Waldhausen categories
ⓘ
define K-theory of exact categories ⓘ |
| usesConstruction |
Q-construction
ⓘ
plus-construction ⓘ |
| usesTool |
classifying spaces
ⓘ
homotopy groups ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Quillen K-theory Description of subject: Quillen K-theory is a sophisticated algebraic K-theory framework defined via higher K-groups of exact or Waldhausen categories, providing deep invariants of rings, schemes, and topological spaces that generalize and extend earlier constructions such as Milnor K-theory.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Hopf algebra (concept named after him)
linked to: Quillen K-theory
linked to: Quillen K-theory