# nLab strict monoidal category

Contents

### Context

#### Monoidal categories

monoidal categories

# Contents

## Definition

A monoidal category is strict if its associator and left/right unitors are identity natural transformations.

Very explicitely, it means that:

###### Definition

A strict monoidal category is a category $\mathcal{C}$ equipped with an object $1 \in \mathcal{C}$ and a bifunctor $\otimes:\mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}$ such that for every objects $A,B,C$ and morphisms $f,g,h$, we have:

• $(A \otimes B) \otimes C = A \otimes (B \otimes C)$
• $1 \otimes A = A$
• $A \otimes 1 = A$
• $(f \otimes g) \otimes h = f \otimes (g \otimes h)$
• $Id_{1} \otimes f = f$
• $f \otimes Id_{1} = f$

Last revised on August 1, 2022 at 18:09:28. See the history of this page for a list of all contributions to it.