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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.

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    $\begingroup$ 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. $\endgroup$
    – Tim Campion
    Commented Mar 28, 2020 at 17:52
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    $\begingroup$ 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"? $\endgroup$
    – Mike Shulman
    Commented Mar 28, 2020 at 21:44
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    $\begingroup$ 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. $\endgroup$
    – Tim Campion
    Commented Mar 29, 2020 at 15:42

1 Answer 1

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i) A monoid

ii) Representations of a quasi-Hopf algebra

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