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Canonical static concept record

Singular cochain complex

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Summary

The cochain complex obtained by applying a coefficient-valued Hom functor to singular chains.

Record metadata

Canonical name: Singular cochain complex
Concept ID: singular_cochain_complex
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Algebraic topology
Domains: Algebraic topology, Homological algebra
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

singular chain complex
coefficient group or ring

Data

cochains and coboundary

Axioms / constraints

$\delta^2=0$

Canonically induces

Singular cohomology

Concept sources

Wikipedia — Singular cohomology — Singular cohomology · encyclopedic reference · source ID wp-singular-cochain-complex
Hatcher — Algebraic Topology — Algebraic Topology · textbook · source ID hatcher-algebraic-topology

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Cochain complex Singular cochain complex

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_cochain_complex_to_singular_cochain_complex_classification
Relation type: Classification classification
Direction: source → target
Endpoint roles: source: Has subtype; target: Classified as
Authored annotation: has singular cochains as a specialization

Authored explanation

The singular cochain complex with coefficients is a particular cochain complex obtained by applying Hom to singular chains.

How to interpret this relation type

The target is a member or subtype of the broader source class.

Relation sources

Hatcher — Algebraic Topology — Algebraic Topology · textbook · source ID hatcher-algebraic-topology

Singular chain complex Singular cochain complex

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_singular_chain_complex_to_singular_cochain_complex_canonical_construction
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: dualize with coefficients

Authored explanation

Applying $\operatorname{Hom}(-,A)$ produces singular cochains.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Hatcher — Algebraic Topology — Algebraic Topology · textbook · source ID hatcher-algebraic-topology

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Each relation below starts at this concept.