# nLab strict monoidal category

Contents

### Context

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# Contents

## Definition

A monoidal category-structure is strict if its associator and left/right unitors are identity natural transformations. By the coherence theorem for monoidal categories, every monoidal category is monoidally equivalent to a strict one.

Explicitly, this 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:

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

## Examples

###### Example

The skeletal version of the symmetric groupoid is the free strict symmetric monoidal category on a single object.

## References

Last revised on February 7, 2024 at 09:38:14. See the history of this page for a list of all contributions to it.