CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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A two-object indiscrete preorder is equivalent but not isomorphic to its one-object poset reflection

Statement refuted

Equivalence of categories does not imply isomorphism of categories.

Facts & Assumptions

Given: The preorder $C=\{0,1\}$ with $x\le y$ for every $x,y$ , and the one-object poset $D=\{*\}$ .

[L1]

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

[L2]

Quasi-inverse functors with natural isomorphisms give an equivalence (Equivalence, quasi-inverse, and adjoint equivalence of categories).

[L3]

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

Counterexample

technique · direct

1.1

Let $F:C\to D$ be the unique functor and let $G:D\to C$ select $0$ . Then $FG=1_D$ .

1.2

But $C$ has two objects and $D$ has one. No functor between them is bijective on objects, so [L3] rules out an isomorphism of categories.

2.1

The functor $GF$ is constant at $0$ . Since $C$ has exactly one arrow between every ordered pair of objects, the unique arrows $x\to0$ are the components of a natural isomorphism $1_C\Rightarrow GF$ .

step 1.1L1

3.1

Hence $F$ and $G$ exhibit $C\simeq D$ by [L2].

step 1.1step 2.1L2

4.1

Thus $C$ and its one-object poset reflection $D$ are equivalent but not isomorphic.

step 3.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 27 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, discussion after Definition 1.5.5 (standard reference, not scraped)