PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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.

Pullbacks of fibrations are fibrations

Statement

Let $p : E \to B$ be a Hurewicz or Serre fibration and $g : A \to B$ continuous. The pullback projection $q : g^{*} E \to A$ , where $g^{*} E = {(a, e) : g (a) = p (e)}$ and $q (a, e) = a$ , is a fibration of the same type. Use ordinary subspaces and products in ordinary spaces, and kified subspaces and k-products in CGWH. No surjectivity or AC is required.

Facts & Assumptions

[F1]

HLP supplies one lift of a compatible homotopy problem. Hurewicz and serre fibrations

[F2]

Pairing continuous maps gives a continuous map into a product, without its separate surjectivity/choice clause. A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice

[F3]

A continuous ambient map landing in a subspace is continuous into that subspace. Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace

Proof

Given: Continuous $v : X \to g^{*} E$ and $H : X \times I \to A$ with $q v = H (-, 0)$ , for an allowed test space $X$ .

1.1F1F2

Write $v = (v_{A}, v_{E})$ . Then $p v_{E} = g v_{A} = g H (-, 0)$ . By F1 the base homotopy $g H$ has a lift $L : X \times I \to E$ with $L (-, 0) = v_{E}$ and $p L = g H$ . This uses exactly the original test class: every space for ordinary Hurewicz, every CGWH space for its CG version, or every disk for Serre.

2.1F2F3step 1.1

Set $\tilde{H} (x, t) = (H (x, t), L (x, t))$ . Its coordinates are continuous, it lands in the pullback by $p L = g H$ , and F2–F3 give continuity. In CGWH this is the same categorical pairing into the kified pullback; the CG source property gives the kified target map. Its initial value is $(v_{A}, v_{E}) = v$ , and $q \tilde{H} = H$ . Thus it solves the required HLP problem.

3.1step 1.1step 2.1∎

Empty parameter spaces have the empty lift; if the pullback is empty, every allowable initial-map problem has empty parameter space. Disk dimension zero is included in step 1.1. Constant base maps, point bases, $t = 0$ and $t = 1$ satisfy the same equations, with no uniqueness or regularity needed. Only one existential HLP witness is used; no indexed selections are made. This proves the claim for both types.

Pullbacks of fibrations are fibrations

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources