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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.

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    $\begingroup$ I believe most people would just call this a "strictly associative monoidal category". $\endgroup$
    – varkor
    Nov 18 '21 at 23:12
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    $\begingroup$ 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. $\endgroup$
    – john
    Nov 19 '21 at 14:14
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Chris Douglas and I invented the term "dicategory" to describe that exact situation: a bicategory where every associator is an identity morphism.

(See our paper: Internal Bicategories)


We needed that term because Conformal Nets form the objects of a (weak) dicategory object in CAT, where CAT stands for the 2-category of categories.

(See our paper: Conformal nets IV: the 3-category)

[In fact, we proved something slightly stronger: conformal nets form the objects of a weak dicategory object in SMC, where SMC stands for the 2-category of symmetric monoidal categories.]

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In Joachim Kock's paper Weak identity arrows in higher categories he defined a "fair 2-category" to be (a nonalgebraic version of) a bicategory with strict associators. So you could call it a "fair monoidal category".

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