LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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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.

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Homotopy-cardinality pushforward is functorial

Statement

For maps $f : X \to Y$ and $g : Y \to Z$ of locally finite homotopy types,

(g \circ f)_{!} = g_{!} \circ f_{!} : Q_{c} (X) ⟶ Q_{c} (Z) .

Facts & Assumptions

Given: Locally finite homotopy types $X, Y, Z$ , maps $f : X \to Y$ , $g : Y \to Z$ , and $α \in Q_{c} (X)$ .

[F1]

The pushforward at a component is the component sum with ratio weight

r_{f} (x, y) : = \prod_{i > 0} ∣ π_{i} (X, x) ∣^{(- 1)^{i}} ∣ π_{i} (Y, y) ∣^{(- 1)^{i + 1}}

for each $x$ mapping to $y$ (Homotopy-fiber weights reduce to componentwise ratios).

Proof

1.1F1algebra

If $x \in π_{0} (X)$ maps to $y \in π_{0} (Y)$ and $y$ maps to $z \in π_{0} (Z)$ , then [F1] gives $r_{f} (x, y) r_{g} (y, z) = r_{g \circ f} (x, z)$ , because the exponents of every $∣ π_{i} (Y, y) ∣$ sum to zero.

2.1F1step 1.1∎

Evaluate $g_{!} f_{!} α$ at $z$ . Its finite sum is over pairs $(x, y)$ with $f (x) = y$ and $g (y) = z$ ; each $x$ determines the unique component $y = f (x)$ . By step 1.1 its weight is $r_{g \circ f} (x, z)$ , exactly the component formula for $(g \circ f)_{!} α (z)$ from [F1]. Hence the two pushforwards agree at every component.

Depends on

Used by

Toën derived Hall multiplication is associative and unital Theorem

Dependency tree · two levels

5 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

Bertrand Toën, Derived Hall algebras, arXiv:math/0501343v4, §2, Corollary 2.4 (standard reference, not scraped)