CorollaryStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Evaluation is epic and coevaluation monic for nonzero objects

Statement

If $X \neq 0$ in a tensor category and $X^{\lor}$ is a left dual, then ${ev}_{X} : X^{\lor} \otimes X \to 1$ is epic and ${coev}_{X} : 1 \to X \otimes X^{\lor}$ is monic.

Facts & Assumptions

Given: A nonzero object $X$ of a tensor category and a left dual $X^{\lor}$ .

[F1]

The unit is simple (The unit object of a tensor category is simple).

Proof

technique · direct

1.1

The zig-zag identity shows that evaluation is nonzero: otherwise its composite giving $1_{X}$ would vanish. Its image is therefore a nonzero subobject of the simple object $1$ , hence all of $1$ by [F1]. Thus evaluation is epic.

F1given

2.1

The other zig-zag identity shows directly that coevaluation is nonzero: if it vanished, its composite giving $1_{X}$ would vanish. Since its source $1$ is simple by [F1], its kernel is either $0$ or $1$ ; the nonzero map excludes the latter. Thus coevaluation is monic.

F1given∎

Depends on

The unit object of a tensor category is simple

Used by

An exact k-linear strong monoidal functor to a nonzero multitensor category is faithful Theorem

Dependency tree · two levels

3 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

Etingof, Gelaki, Nikshych, Ostrik, Tensor Categories, Corollary 4.3.9 (standard reference, not scraped)
Etingof, Gelaki, Nikshych, Ostrik, Corrections to Tensor Categories, Chapter 4 (standard reference, not scraped)