DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

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.

Codensity monad

Definition

Let $G : B \to C$ be a functor, and suppose a right Kan extension of $G$ along itself is supplied:

$ε : T G \Rightarrow G$

with $T : C \to C$ (Left and right Kan extensions, Natural transformation and its components).

The codensity monad of $G$ is the triple $(T, η, μ)$ on $C$ , where the underlying endofunctor is $T$ , the unit $η : 1_{C} \Rightarrow T$ is the unique natural transformation with

$1_{G} = ε \circ (η G),$

and the multiplication $μ : T^{2} \Rightarrow T$ is the unique natural transformation with

$ε \circ (μ G) = ε \circ (T ε) .$

The theorem The codensity construction satisfies the monad laws ↗ proves that these data satisfy the axioms of Monad on a category.

Depends on

Used by

Dependency tree · two levels

6 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, 2nd ed., Definition 6.5.12 (standard reference, not scraped)
T. Leinster, Codensity and the ultrafilter monad, §2 (standard reference, not scraped)