LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-04 (gpt-6.1-sol)

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.

Fully faithful functor with thickly generating image is an equivalence

Statement

Assume AC in the ambient set universe used for the category presentations. Let $A$ and $B$ be triangulated categories presented there by sets of objects and morphisms, with $A$ idempotent-complete (every idempotent endomorphism splits). Let $F : A \to B$ be an exact fully faithful functor. If the smallest full triangulated subcategory of $B$ that contains every object isomorphic to some $F (a)$ and is closed under direct summands is all of $B$ , then $F$ is an equivalence.

Without set-size or choice hypotheses, the same thick-image argument still proves that an exact fully faithful functor with the stated idempotent-complete source and thickly generating image is essentially surjective. For definable class presentations, if an objectwise definable class section $(G (b), u_{b} : F (G (b)) \tilde{\to} b)$ is supplied explicitly, the same Hom-lifting formula gives a class quasi-inverse; no class-choice axiom is inferred.

Here an exact functor preserves the translation and distinguished triangles; fully faithful means bijective on every morphism set; and the stated smallest subcategory is the thick closure of the essential image.

Statement references

Paul Seidel, Homological mirror symmetry for the quartic surface, arXiv:math/0310414v4, §2b, pp. 8–9, especially Lemma 2.5 and its proof. That lemma gives the split-closed derived-category instance; the present abstract triangulated-category formulation uses the same essential-image closure argument, proved locally below.

The Axiom of Choice

Facts & Assumptions

Given: The triangulated structures, exactness and ambient set presentations are as stated; use AC in that ambient set universe.

[F1]

A triangulated category consists of a category, a translation, and a specified class of distinguished triangles (Triangulated category).

[F2]

The Axiom of Choice supplies a simultaneous choice from a set-indexed family of nonempty sets (The Axiom of Choice). It is used only for the objectwise witnesses in step 3.1, not as a proper-class choice principle.

Proof

1.1givenF1def. triangulated subcategory

Let $Im (F)$ be the full subcategory of objects isomorphic to some $F (a)$ . Exactness makes it closed under translations. For a morphism between two such objects, full faithfulness lifts the morphism to $A$ ; exactness sends a cone triangle for that lift to a distinguished triangle in $B$ . The cone is therefore again in $Im (F)$ . Thus the image is triangulated.

2.1givenstep 1.1algebra

Suppose $b$ is a direct summand of $F (a)$ , with inclusion $i : b \to F (a)$ and retraction $r : F (a) \to b$ , so $e = i r$ is idempotent. Fullness gives $e = F (e_{a})$ for some endomorphism $e_{a}$ of $a$ , and faithfulness gives $e_{a}^{2} = e_{a}$ . Idempotent completeness of $A$ splits $e_{a}$ ; applying $F$ exhibits $b$ as isomorphic to the image of that splitting. Hence $Im (F)$ is closed under direct summands.

3.1givenF2step 2.1constructalgebra∎

The image is therefore thick. By the assumed thick-generation condition it is all of $B$ up to isomorphism, so $F$ is essentially surjective. For every $b \in Ob (B)$ the set $W_{b} = {(a, u) : a \in Ob (A), u : F (a) \tilde{\to} b}$ is nonempty; it is a set because the chosen object and morphism collections are sets. By [F2] choose $(G (b), u_{b}) \in W_{b}$ simultaneously. For $f : b \to b^{'}$ , let $G (f)$ be the unique morphism with $F (G (f)) = u_{b^{'}}^{- 1} f u_{b}$ . The Hom bijections prove $G (1) = 1$ and $G (g f) = G (g) G (f)$ . The $u_{b}$ give a natural isomorphism $F G \Rightarrow 1_{B}$ . Full faithfulness uniquely lifts $u_{F (a)}^{- 1} : F (a) \to F G F (a)$ to an isomorphism $η_{a} : a \to G F (a)$ ; its naturality follows by applying the faithful $F$ to the required square and using the naturality of $u$ . Thus $1_{A} \Rightarrow G F$ is also a natural isomorphism, and $G$ is a quasi-inverse. The essential-surjectivity part of this step and the closure arguments of steps 1.1–2.1 use no choice or object-set bound. If the displayed objectwise lifts are instead supplied as definable class data, the unique Hom lifts define a class functor by the same formulas and the same naturality verification. This constructs both asserted variants and invokes no global class choice.

Definition references

Triangulated category

Depends on

Dependency tree · 2 levels

Sources

Paul Seidel, Homological mirror symmetry for the quartic surface, arXiv:math/0310414v4