PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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.

Equivalence of categories is reflexive, symmetric, and transitive

Statement

Equivalence of categories is reflexive, symmetric, and transitive.

Facts & Assumptions

Given: Categories and equivalence data between them.

[L1]

An equivalence consists of quasi-inverse functors and natural isomorphisms in both composite directions (Equivalence, quasi-inverse, and adjoint equivalence of categories).

[L2]

Vertical composition is componentwise (Identity natural transformation and vertical composition); whiskering and horizontal composition produce natural transformations between composite functors (Whiskering and horizontal composition of natural transformations), and those horizontal composites are natural (Horizontal composites of natural transformations satisfy naturality).

Proof

technique · direct

1.1

The identity functor is its own quasi-inverse and the identity natural transformations supply an equivalence $C ≃ C$ , proving reflexivity.

givenL1

2.1

If $(F, G, η, ε)$ gives $C ≃ D$ , then $(G, F, ε^{- 1}, η^{- 1})$ gives $D ≃ C$ , proving symmetry.

step 1.1L1

3.1

If $(F, G)$ gives $C ≃ D$ and $(H, K)$ gives $D ≃ E$ , then $H F$ and $G K$ are quasi-inverses; whiskering and vertically composing the two units gives $1_{C} \Rightarrow G K H F$ , and doing the same with the counits gives $H F G K \Rightarrow 1_{E}$ , so [L2] proves transitivity.

step 2.1L1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)
Ahrens, Kapulkin and Shulman, Univalent categories and the Rezk completion, section 6 (standard reference, not scraped)