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

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A preorder is a category with at most one morphism between any two objects, and its functors are exactly monotone maps

Statement

A preorder determines a category with at most one morphism between any two objects, and functors between such categories are exactly monotone maps.

Facts & Assumptions

Given: Preorders $(P,\le_P)$ and $(Q,\le_Q)$ .

[L1]

A preorder is reflexive and transitive, and a monotone map preserves its relation (Preorder and monotone map).

[L2]

Category identities and composition have the meanings of Category, object, morphism, domain, codomain, identity, composition, and hom-collection. In this proposition, a functor is an assignment on objects and arrows that preserves identities and composition.

Proof

technique · direct

1.1

Make the elements of $P$ objects and put one morphism $x\to y$ when $x\le_P y$ , and none otherwise; reflexivity supplies identities and transitivity supplies the unique possible composites, so [L1] gives a category.

givenL1L2

2.1

A function $F:P\to Q$ extends to a functor exactly when $x\le_P y$ guarantees a target morphism $F(x)\to F(y)$ , which is exactly $F(x)\le_Q F(y)$ .

step 1.1L1L2

3.1

Thus the functors between the associated categories are precisely the monotone maps, with no extra arrow choices because every relevant hom-collection has at most one member.

step 2.1L1∎

Depends on

Used by

A two-object indiscrete preorder is equivalent but not isomorphic to its one-object poset reflection Counterexample
Every equivalence of categories is an isomorphism of categories False statement

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 13 results over 5 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, Chapter 1 (standard reference, not scraped)