DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

Comonad on a category

Definition

Let $C$ be a category. A comonad on $C$ is a triple $(G, ε, δ)$ consisting of an endofunctor $G : C \to C$ , a counit $ε : G \Rightarrow 1_{C}$ , and a comultiplication $δ : G \Rightarrow G^{2}$ such that

$G δ \circ δ = δ G \circ δ : G \Rightarrow G^{3},$

$G ε \circ δ = 1_{G} = ε G \circ δ : G \Rightarrow G .$

Equivalently, a comonad on $C$ is a monad (Monad on a category) on the opposite category $C^{o p}$ (Opposite category $C^{o p}$ ), with every arrow reversed.

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 5.1.6 (standard reference, not scraped)