CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The smooth and continuous homotopy categories of smooth manifolds have the same morphism sets

Statement

For smooth manifolds $M$ and $N$ , the set of smooth-homotopy classes of smooth maps $M \to N$ is naturally the same as the set of ordinary homotopy classes of continuous maps $M \to N$ .

Facts & Assumptions

Given: Smooth manifolds $M$ and $N$ .

[L1]

Every continuous map is homotopic to a smooth map (Every continuous map between smooth manifolds is homotopic to a smooth map).

[L2]

Continuous homotopies between smooth maps can be smoothed (Continuously homotopic smooth maps are smoothly homotopic).

Proof

technique · direct

1.1

By [L1], every continuous homotopy class has at least one smooth representative.

L1given

1.2

If two smooth maps are homotopic as continuous maps, then [L2] upgrades that continuous homotopy to a smooth one. Thus two smooth representatives lie in the same smooth-homotopy class exactly when they lie in the same continuous homotopy class.

L2given

2.1

Steps 1.1 and 1.2 identify the two morphism sets canonically.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Smooth Approximation of Maps Between Manifolds (standard reference, not scraped)