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Canonical static concept record

Homotopy

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Summary

A continuous deformation between two continuous maps, represented by a map $H:X\times[0,1]\to Y$ .

Record metadata

Canonical name: Homotopy
Concept ID: homotopy
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Algebraic topology
Domains: Algebraic topology, Topology
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

spaces $X,Y$
maps $f,g:X\to Y$

Data

continuous family $H_t$

Axioms / constraints

endpoint conditions $H_0=f$ and $H_1=g$

Canonically induces

An equivalence relation on maps with fixed domain and codomain

Concept sources

Wikipedia — Homotopy — Homotopy · encyclopedic reference · source ID wp-homotopy
nLab — Homotopy — Homotopy · mathematical reference · source ID nlab-homotopy
Hatcher — Algebraic Topology — Algebraic Topology · textbook · source ID hatcher-algebraic-topology

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Homotopy equivalence Homotopy

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

Relation ID: story_homotopy_equivalence_to_homotopy_induced_forgotten
Relation type: Induced / forgotten induced-forgotten
Direction: source → target
Endpoint roles: source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation: is witnessed by homotopies

Authored explanation

A homotopy equivalence consists of maps whose composites are homotopic to the relevant identity maps; retaining a witnessing deformation gives homotopy data.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

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

Topological space Homotopy

Permalink to relation

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

Relation ID: story_topological_space_to_homotopy_add_data
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: deform maps continuously

Authored explanation

A homotopy is a continuous one-parameter family of maps.

How to interpret this relation type

Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.

Relation sources

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

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Each relation below starts at this concept.

No direct relations are authored in this direction.

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Homotopy

Summary

Record metadata

Carrier(s)

Data

Axioms / constraints

Canonically induces

Concept sources

Incoming relations (arrows to this concept)

Homotopy equivalence Homotopy

Authored explanation

How to interpret this relation type

Relation sources

Topological space Homotopy

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)