TheoremStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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The monoid-object axioms may be written without associators

Statement

Let $(M, μ, η)$ be a monoid object in a monoidal category. Then, after the coherence identification of unbracketed tensor strings, its axioms may be written as $μ \circ (μ \otimes 1_{M}) = μ \circ (1_{M} \otimes μ),$ $μ \circ (η \otimes 1_{M}) = 1_{M}, μ \circ (1_{M} \otimes η) = 1_{M},$ without displaying associators or unitors.

Facts & Assumptions

Given: A monoid object $(M, μ, η)$ in a monoidal category $C$ .

[L1]

A monoid object is defined by the associativity equation $μ \circ (μ \otimes 1_{M}) = μ \circ (1_{M} \otimes μ) \circ α_{M, M, M}$ and the two unit equations with $λ_{M}$ and $ρ_{M}$ (Monoid objects and comonoid objects in a monoidal category).

[L2]

The category $C$ is monoidally equivalent to a strict monoidal category (Mac Lane strictification).

[L3]

Unbracketed tensor strings are well defined after coherence (Unbracketed tensor strings are well defined after coherence).

Proof

technique · direct

1.1

Choose a strictification of $C$ as in [L2]. Applying the strong monoidal functor of that equivalence to the diagrams from [L1] transports the monoid-object structure of $M$ to a monoid-object structure on its image in a strict monoidal category.

L1L2choose

2.1

In the strict target, the associator and unitors are identities, so the transported axioms become exactly the displayed unbracketed equalities.

L2step 1.1

3.1

By [L3], those unbracketed composites denote definite maps back in the original category and do not depend on which brackets were suppressed. Hence the monoid-object axioms may be written in the simplified form without changing their content.

L3step 2.1∎

Depends on

Used by

A monoid object written with and without associators Example

Dependency tree · two levels

13 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, Chapter 2.7 (standard reference, not scraped)