CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-13

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Assuming Choice, every small complete category and every small cocomplete category is a preorder

Statement

Assume Choice. Every small complete category is a preorder, and every small cocomplete category is a preorder.

Facts & Assumptions

Given: A small category $C$ that is complete or cocomplete.

[L1]

If a small category has constant products or coproducts indexed by the cardinality of its morphism set, it is a preorder (Assuming Choice, a small category with products or coproducts indexed by the cardinality of its morphism set is a preorder).

[F1]

Complete categories have every small limit, and cocomplete categories have every small colimit (Finite, small, and large limits and colimits; complete and cocomplete categories).

Proof

technique · direct corollary

1.1

Let $κ = ∣ Mor (C) ∣$ . Since $C$ is small, a discrete category on the set $κ$ is a small indexing category.

given

2.1

If $C$ is complete, [F1] supplies the product of every constant $κ$ -family. The product clause of [L1] makes $C$ a preorder.

F1L1step 1.1

3.1

If $C$ is cocomplete, [F1] instead supplies every constant $κ$ -coproduct, and the coproduct clause of [L1] gives the same conclusion.

F1L1step 1.1∎

Depends on

Used by

Why completeness alone cannot replace a solution set or the SAFT smallness hypotheses Remark

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

E. Riehl, Category Theory in Context, Corollary after Proposition 3.7.3 (standard reference, not scraped)