# Examples of strict monoidal categories and monoidal categories with nontrivial associators

What are some "natural" motivating examples of the following:

i) A strict monoidal category,

ii) A monoidal with non-trivial associatots?

For i) the only examples I know are categories which have been strictified, are there any examples occuring "in nature" which are strict, or is strictness in some sense an "unnatural" or artificial requirement?

For ii) I should clarify what I mean by "non-trivial" - basically the examples I consider trivial are tensor products of vector spaces, bimodules, representations, and so on, where the associator is just the elementary rewritting of brackets.

• Regarding (i), every monoidal category is monoidally equivalent to a strict monoidal category. An example of a "naturally occurring" strict monoidal category is the category $\Delta_+$ of finite ordinals, under ordinal sum. This is the free strict monoidal category on a monoid object. I find (ii) more interesting -- in particular, I believe there are interesting examples of categories equipped with a tensor bifunctor which admit more than one possible associator, and the resulting monoidal structures are not equivalent. I hope somebody can provide such examples. Mar 28, 2020 at 17:52
• It's hard to tell what you mean by "non-trivial" if you don't know any examples. One might go so far as to claim that the associator always consists of rewriting of brackets, essentially by definition. How about a 2-group constructed from a group cohomology class; would you regard its associator as always "trivial"? Mar 28, 2020 at 21:44
• In this paper Mark Hovey studies the problem of classifying symmetric monoidal closed structure on the category $Mod_R$ of $R$-modules for a ring $R$. For example, when $R$ is a field, there is a unique such, and when $R = \mathbb F_2[C_2]$ there are exactly 7 up to equivalence. He doesn't cleanly separate out the problem of understanding non-symmetric monoidal structures, but I think his analysis should probably be enlightening. Mar 29, 2020 at 15:42