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

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Strict monoidal category

Definition

A strict monoidal category is a monoidal category $(C, \otimes, 1, α, λ, ρ)$ (Monoidal category) in which

$((X \otimes Y) \otimes Z) = X \otimes (Y \otimes Z), 1 \otimes X = X, X \otimes 1 = X$

as literal equalities of objects, and the constraint isomorphisms are identities:

$α_{X, Y, Z} = 1_{X \otimes (Y \otimes Z)}, λ_{X} = 1_{X}, ρ_{X} = 1_{X} .$

Thus a strict monoidal category does not merely identify the two bracketings up to a specified isomorphism; it makes them equal on the nose.

Depends on

Monoidal category

Used by

Two formally distinct words can become the same object Counterexample
The braid category Definition
FALSE: every monoidal category is strict False statement
A monoidal category equivalent to a strict one satisfies coherence Theorem
A poset with finite meets is a strict monoidal category Theorem
In a strict braided monoidal category the braiding satisfies the Yang-Baxter equation Theorem
The endofunctor category of a small category is strict monoidal under composition Theorem
The right-module endofunctor category is strict monoidal Theorem
The two-braid category is strict rigid monoidal Theorem

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

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Definition 2.8.1 (standard reference, not scraped)