LemmaStatement: AI-generatedProof: AI-generated

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.

Cofiber of a direct-summand inclusion

Statement

For spectra $A$ and $B$ , the cofiber of the canonical inclusion $A \to A \oplus B$ is naturally equivalent to $B$ . Under this equivalence the cofiber map is the projection $A \oplus B \to B$ , and the connecting map $B \to A [1]$ is zero. Here $\oplus$ denotes the finite coproduct, which is also a product in the stable category.

Facts & Assumptions

Given: Spectra $A$ and $B$ and their canonical inclusion and projection.

Goal: Identify the cofiber sequence and its connecting map.

[F1]

Stable spectra and exact triangles supplies the stable category of spectra, zero object, cofibers, and suspension.

Proof

1.1F1givenconstruct

The pushout defining $cofib (A \to A \oplus B)$ is $(A \oplus B) ⨿_{A} 0$ . By the universal property of coproducts and the zero object, maps from this pushout to any spectrum are naturally maps from $B$ ; hence the pushout is naturally $B$ , with cofiber map the projection.

2.1F1step 1.1∎

The projection has the canonical section $B \to A \oplus B$ . In the cofiber triangle, the composite of consecutive maps is zero, so the connecting map composed with this projection is zero. Composing with the section shows that the connecting map itself is zero.

Depends on

Stable spectra and exact triangles

Used by · 2

Dependency tree · 3 levels

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