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Canonical static concept record

Loop space $\Omega X$

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Summary

The space of based loops in a pointed space, with concatenation associative up to coherent homotopy.

Record metadata

Canonical name: Loop space $\Omega X$
Concept ID: loop_space
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)

pointed space $(X,x_0)$

Data

based continuous maps $S^1\to X$

Canonically induces

Homotopy groups through path components and iteration

Concept sources

Wikipedia — Loop space — Loop space · encyclopedic reference · source ID wp-loop-space
Hatcher — Algebraic Topology — Algebraic Topology · textbook · source ID hatcher-algebraic-topology

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Pointed topological space Loop space $\Omega X$

Permalink to relation

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

Relation ID: story_pointed_topological_space_to_loop_space_canonical_construction
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: form based loops

Authored explanation

The loop space consists of maps from the pointed circle preserving the basepoint.

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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