DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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.

The ultrafilter endofunctor with principal unit and flattening multiplication

Definition

The ultrafilter endofunctor $β : S e t \to S e t$ sends $X$ to the set $β X$ of ultrafilters on $X$ and sends $f : X \to Y$ to ultrafilter pushforward $f_{*}$ . Its principal unit and flattening multiplication are

$η_{X} (x) = {A \subseteq X : x \in A},$

$μ_{X} (W) = {A \subseteq X : {U \in β X : A \in U} \in W} .$

The preceding pushforward and flattening lemmas establish that these assignments are well-defined and natural.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 6 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, 2nd ed., Example 5.1.4(v) and Exercise 5.1.ii (standard reference, not scraped)