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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:

\begin{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:

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

\end{definition}

Examples

\begin{example} The skeletal version of the symmetric groupoid is the free strict symmetric monoidal category on a single object. \end{example}

Related concepts

• coherence theorem for monoidal categories

• permutative category

• commutative monoidal category

References

• Saunders MacLane, §XI.3 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

• Peter Schauenburg, Turning Monoidal Categories into Strict Ones, New York Journal of Mathematics 7 (2001) 257-265 [[nyjm:j/2001/7-16](http://nyjm.albany.edu/j/2001/7-16.html), eudml:121925]