False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-11

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Every equivalence of categories is an isomorphism of categories

Statement

FALSE. Every equivalence of categories is an isomorphism of categories.

Facts & Assumptions

Given: The indiscrete preorder $C = {0, 1}$ and the terminal one-object category $D = {*}$ .

[L1]

Preorders define thin categories (A preorder is a category with at most one morphism between any two objects, and its functors are exactly monotone maps).

[L2]

Equivalence is witnessed by quasi-inverses up to natural isomorphism (Equivalence, quasi-inverse, and adjoint equivalence of categories).

[L3]

Isomorphism of categories requires a bijection on objects (A functor is an isomorphism of categories exactly when its object and morphism maps are bijective).

Refutation

technique · direct

1.1

The unique $F : C \to D$ and the functor $G : D \to C$ selecting $0$ satisfy $F G = 1_{D}$ . The unique arrows $x \to 0$ in the indiscrete preorder form a natural isomorphism $1_{C} \Rightarrow G F$ .

1.2

No functor $C \to D$ is bijective on objects because $C$ has two objects and $D$ one, so the categories are not isomorphic by [L3].

2.1

Therefore $F$ is an equivalence by [L2].

step 1.1L2

3.1

This equivalent but nonisomorphic pair refutes the statement.

step 2.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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, discussion after Definition 1.5.5 (standard reference, not scraped)