DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-13‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Filtered categories and filtered colimits

Definition

A category $J$ is filtered when every finite diagram in $J$ admits a cocone. Equivalently:

$J$ is nonempty;
for every $j, k$ there are an object $ℓ$ and arrows $j \to ℓ \leftarrow k$ ;
for every parallel pair $f, g : j ⇉ k$ , there is $h : k \to ℓ$ with $h f = h g$ .

The equivalence uses finite diagrams as in Finite, small, and large limits and colimits; complete and cocomplete categories; the nonempty and zigzag language agrees with the connectedness convention in Isomorphism, groupoid, and connected category. A filtered colimit is a colimit (Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties) of a diagram indexed by a small filtered category. Smallness is included so the colimit is among the ordinary small colimits.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 17 results over 7 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

E. Riehl, Category Theory in Context, Definition 3.8.7 (standard reference, not scraped)