# "Partially strict" monoidal categories

Recall that a monoidal category $$\newcommand{\C}{\mathcal{C}} (\C,\otimes,I)$$ comes equipped with an associator $$\alpha_{XYZ} \colon X \otimes (Y \otimes Z) \xrightarrow{\sim} (X \otimes Y) \otimes Z$$ as well as left and right unitors $$\lambda_X \colon I \otimes X \xrightarrow{\sim} X$$ and $$\rho_X \colon X \otimes I \xrightarrow{\sim} X$$ (where $$X$$, $$Y$$, and $$Z$$ are objects of $$\C$$). Then $$\C$$ is said to be strict if the associator and left and right unitors are all identity morphisms.

Question: Is there a term for the situation where the associator is the identity, but where the unitors need not be identities?

The reason I ask is because I typically treat tensor products of $$R$$-modules (where $$R$$ is a commutative ring) in this way. I think of there being a uniquely determined triple tensor product $$L \otimes_R M \otimes_R N$$ so that there is essentially no difference between the two different ways to associate the triple tensor. However, I tend to think of $$R \otimes_R M$$ as being only naturally isomorphic to $$M$$, not equal to it.

• I believe most people would just call this a "strictly associative monoidal category". Nov 18 '21 at 23:12
• On a different but analogous topic --- a skew monoidal category is called Hopf" if the associator $(XY)Z \to X(YZ)$ is invertible, because in the case of the skew monoidal structure induced by a bialgebra, the associator is invertible just when the original bialgebra is Hopf.
– john
Nov 19 '21 at 14:14