DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 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.

Preorder and monotone map

Definition

A preorder on a set $P$ is a relation $\le$ that is reflexive and transitive. Unlike a partial order (Partial order and partially ordered set), it need not be antisymmetric.

A map $f:P\to Q$ between preorders is monotone when $x\le_P y$ implies $f(x)\le_Q f(y)$ . Every partial order is a preorder, and the definition of monotone map agrees with the usual one for partially ordered sets.

Depends on

Partial order and partially ordered set

Used by

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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 1 result over 1 level. 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)