Triple

T10829222
Position Surface form Disambiguated ID Type / Status
Subject J. H. C. Whitehead E255572 entity
Predicate knownFor P22 FINISHED
Object Whitehead product in homotopy theory
The Whitehead product in homotopy theory is a bilinear operation on homotopy groups that captures how spheres can be nontrivially linked or composed within a topological space.
E886920 NE FINISHED

How this triple was built (4 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Whitehead product in homotopy theory | Statement: [J. H. C. Whitehead, knownFor, Whitehead product in homotopy theory]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Whitehead product in homotopy theory
Context triple: [J. H. C. Whitehead, knownFor, Whitehead product in homotopy theory]
  • A. Eilenberg–MacLane spaces
    Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of cohomology.
  • B. Classifying Spaces and Fibrations
    "Classifying Spaces and Fibrations" is a mathematical work that develops the theory of classifying spaces in algebraic topology and their relationship to fiber bundles and fibrations.
  • C. Sullivan minimal model in rational homotopy theory
    The Sullivan minimal model in rational homotopy theory is a canonical commutative differential graded algebra that encodes the rational homotopy type of a topological space in an algebraic form.
  • D. Whitney approximation theorem
    The Whitney approximation theorem is a fundamental result in differential topology stating that any continuous function between smooth manifolds can be uniformly approximated by smooth functions.
  • E. "Algebraic Topology"
    "Algebraic Topology" is a foundational mathematical text that develops topological concepts using algebraic methods such as homology and cohomology theories.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Whitehead product in homotopy theory
Triple: [J. H. C. Whitehead, knownFor, Whitehead product in homotopy theory]
Generated description
The Whitehead product in homotopy theory is a bilinear operation on homotopy groups that captures how spheres can be nontrivially linked or composed within a topological space.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Whitehead product in homotopy theory
Target entity description: The Whitehead product in homotopy theory is a bilinear operation on homotopy groups that captures how spheres can be nontrivially linked or composed within a topological space.
  • A. Eilenberg–MacLane spaces
    Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of cohomology.
  • B. Classifying Spaces and Fibrations
    "Classifying Spaces and Fibrations" is a mathematical work that develops the theory of classifying spaces in algebraic topology and their relationship to fiber bundles and fibrations.
  • C. Sullivan minimal model in rational homotopy theory
    The Sullivan minimal model in rational homotopy theory is a canonical commutative differential graded algebra that encodes the rational homotopy type of a topological space in an algebraic form.
  • D. Whitney approximation theorem
    The Whitney approximation theorem is a fundamental result in differential topology stating that any continuous function between smooth manifolds can be uniformly approximated by smooth functions.
  • E. "Algebraic Topology"
    "Algebraic Topology" is a foundational mathematical text that develops topological concepts using algebraic methods such as homology and cohomology theories.
  • F. None of above. chosen

Provenance (5 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69d6aa8081448190a9324184f2bd1c26 completed April 8, 2026, 7:20 p.m.
NER Named-entity recognition batch_69d734d3eab88190b30a3025b6b2b0bc completed April 9, 2026, 5:10 a.m.
NED1 Entity disambiguation (via context triple) batch_69de85a068b08190948c3ca32cdda147 completed April 14, 2026, 6:21 p.m.
NEDg Description generation batch_69de8956d9f081909d076c5e413c1f74 completed April 14, 2026, 6:37 p.m.
NED2 Entity disambiguation (via description) batch_69de8e80fe80819088ac76bb5abc58f0 completed April 14, 2026, 6:59 p.m.
Created at: April 8, 2026, 9:19 p.m.