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Canonical static concept record

Singular chain complex

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Summary

The chain complex freely generated in degree $n$ by continuous maps from the standard $n$ -simplex into a space, with alternating face boundary.

Record metadata

Canonical name: Singular chain complex
Concept ID: singular_chain_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)

topological space $X$

Data

free abelian groups $C_n(X)$
boundary maps

Axioms / constraints

$\partial^2=0$

Canonically induces

Singular homology

Concept sources

Wikipedia — Singular homology — Singular homology · encyclopedic reference · source ID wp-singular-chain-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.

Chain complex Singular chain complex

Permalink to relation

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

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

Authored explanation

The singular chain complex of a topological space is a particular chain complex, freely generated by singular simplices with the alternating-face boundary.

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

Topological space Singular chain complex

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

Relation ID: story_topological_space_to_singular_chain_complex_canonical_construction
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: take singular simplices

Authored explanation

Continuous simplices and alternating face maps form the singular chain complex.

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.