DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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.

Monoid objects and comonoid objects in a monoidal category

Definition

Let $(C, \otimes, 1, α, λ, ρ)$ be a monoidal category (Monoidal category).

A monoid object in $C$ is an object $M$ together with morphisms

$μ : M \otimes M ⟶ M, η : 1 ⟶ M,$

such that

$μ \circ (μ \otimes 1_{M}) = μ \circ (1_{M} \otimes μ) \circ α_{M, M, M},$

$μ \circ (η \otimes 1_{M}) = λ_{M}, μ \circ (1_{M} \otimes η) = ρ_{M} .$

A comonoid object in $C$ is an object $C$ together with morphisms

$Δ : C ⟶ C \otimes C, ε : C ⟶ 1,$

such that

$(α_{C, C, C}) \circ (Δ \otimes 1_{C}) \circ Δ = (1_{C} \otimes Δ) \circ Δ,$

$(ε \otimes 1_{C}) \circ Δ = λ_{C}^{- 1}, (1_{C} \otimes ε) \circ Δ = ρ_{C}^{- 1} .$

The associator and unitors are written explicitly because this page does not yet identify different bracketings automatically.

Depends on

Monoidal category

Used by

Dependency tree · two levels

4 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

S. Mac Lane, Categories for the Working Mathematician, Chapter VII.3 (standard reference, not scraped)
P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Definition 2.7.1 (standard reference, not scraped)