DefinitionDefinition: Literature-sourcedProof: AI-generatedprecheck passjudge 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.

Modules over a monoid object, their morphisms, and their category

Definition

Let $M$ be a monoid object in a monoidal category $(C, \otimes, 1, α, λ, ρ)$ (Monoid objects and comonoid objects in a monoidal category).

A left $M$ -module is an object $X$ together with an action morphism

$a : M \otimes X ⟶ X$

such that

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

$a \circ (η \otimes 1_{X}) = λ_{X} .$

A morphism of left $M$ -modules from $(X, a)$ to $(Y, b)$ is a morphism $f : X \to Y$ in $C$ such that

$f \circ a = b \circ (1_{M} \otimes f) .$

Facts & Assumptions

Given: A monoid object $(M, μ, η)$ and left $M$ -modules $(X, a)$ , $(Y, b)$ , and $(Z, c)$ .

[L1]

A category has identities and associative composition (Category, object, morphism, domain, codomain, identity, composition, and hom-collection).

Verification

technique · direct

1.1givenL1

For every module $(X, a)$ , the identity $1_{X}$ is a module morphism because $1_{X} \circ a = a = a \circ (1_{M} \otimes 1_{X})$ . Thus identities exist.

1.2givenL1algebra

If $f : (X, a) \to (Y, b)$ and $g : (Y, b) \to (Z, c)$ are module morphisms, then $(g \circ f) \circ a = g \circ (f \circ a) = g \circ b \circ (1_{M} \otimes f) = c \circ (1_{M} \otimes g) \circ (1_{M} \otimes f) = c \circ (1_{M} \otimes (g \circ f))$ , so $g \circ f$ is again a module morphism.

2.1step 1.1step 1.2L1∎

Because composition in $C$ is associative by [L1], the module morphisms with these identities and composites form a category.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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.4 (standard reference, not scraped)
P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Chapter 2 (standard reference, not scraped)