PropositionStatement: Literature-sourcedProof: 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.
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.

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Posets and monotone maps form the large locally small category $P o s e t$

Statement

Partially ordered sets and monotone maps form a large locally small category $P o s e t$ .

Facts & Assumptions

Given: Posets $P, Q, R$ and monotone maps $f : P \to Q$ , $g : Q \to R$ .

[L1]

Partial orders are reflexive, antisymmetric, and transitive (Partial order and partially ordered set); a composite of monotone maps is monotone by transitivity of implication.

[L2]

The functions $A \to B$ between fixed sets form the set $B^{A}$ (The set $B^{A}$ of all functions $A \to B$ ); category size is as in Category, object, morphism, domain, codomain, identity, composition, and hom-collection and Small, locally small, and large categories, and the ordinals form a proper class (Burali-Forti: there is no set of all ordinals).

Proof

technique · direct

1.1

Identity maps are monotone, and $x \leq y$ gives $f (x) \leq f (y)$ and then $g (f (x)) \leq g (f (y))$ , so $g \circ f$ is monotone; composition is associative and unital as function composition.

givenL1

2.1

Hence posets and monotone maps form a category whose hom-collections are sets of functions.

step 1.1L2

3.1

Each singleton ${α}$ has its equality order, and ordinals yield a proper class of distinct such posets by [L2]; therefore $P o s e t$ is large and locally small.

step 2.1L1L2∎

Depends on

Used by

Open-set and closed-set functors on Topᵒᵖ are naturally isomorphic by complements Example

Dependency tree · two levels

11 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)

Posets and monotone maps form the large locally small category PosetPoset