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I am looking for interesting examples of categories admitting multiple monoidally inequivalent closed (or compact closed) symmetric monoidal structures. We know how to construct disconnected toy models [1], but none of direct practical interest. I also seem to recall from [2] that the category of $G$-sets admits two inequivalent such structures, but I am curious to know whether this is a sporadic occurrence or whether there is a large family of interesting examples lurking somewhere out there.

[1]: see e.g. https://arxiv.org/abs/1803.00708

[2]: Borceux, Francis. Handbook of Categorical Algebra

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3 Answers 3

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A pretty interesting class of examples comes about by classifying compact monoidal groupoids. Given a group $G$, a $G$-module $M$, and a (normalized) 3-cocycle $a: G \times G \times G \to M$, one can manufacture a compact monoidal groupoid whose category of objects is $G$, whose morphisms are ordered pairs $(g, m) \in G \times M$ where we define $\text{dom}(g, m) = \text{cod}(g, m) = g$ and where endomorphism composition is defined by addition in $M$, and where we define the tensor product by $(g, m) \otimes (h, n) = (g h, m + g n)$. It's the 3-cocycle that furnishes the associativity data, and we get monoidally inequivalent groupoids whenever the 3-cocycles are not cohomologous. This was observed by Joyal and Street in their paper Braided Monoidal Categories.

Now you were asking about the symmetric monoidal case (where we now assume $G$ is abelian). These are also known as Picard groupoids. In the simplified scenario where we demand strict associativies and consider only the case where $G$ acts trivially on $M$, in which case the underlying category becomes the product $K G \times B M$ of the evident discrete monoidal category $K G$ with the evident one-object category $B M$, any symmetric bilinear pairing $G \otimes G \to M$ can be used to manufacture a symmetry isomorphism for a symmetric monoidal structure on $K G \times B M$, and these examples are generally symmetric-monoidally inequivalent. I can't tell how far away such examples are from the "toy"examples" you have in mind, but Picard groupoids are surely of interest -- see for example applications to 2-stage Postnikov systems of spectra here.

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The category $Cat$ of small categories admits two inequivalent tensor products: the Cartesian product and the funny product. This generalizes up to 2-cat, 3-cat, etc. By the way, a word of advise to those like me with a murky memory: googling "cat funny product" is going to just give you pictures of cats.

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    $\begingroup$ But he asked for compactness. $\endgroup$
    – Todd Trimble
    Sep 1, 2018 at 18:01
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    $\begingroup$ Omega-cat admits the Crans-Gray-Steiner lax tensor product. I dunno if this meets the compactness requirement, but it is in the same vein. $\endgroup$ Sep 1, 2018 at 18:10
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    $\begingroup$ At present, the question asks for compact or compact closed, but anyway, I think all the answers are pointing the OP towards examples from category theory, and that's the important thing. $\endgroup$ Sep 1, 2018 at 18:39
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The large class of examples I have in mind, though I am not sure if it meets your compactness requirement (definition?) are the Lax tensor products on $n$-Cat. It has been constructed in the cases $n=2$ and $n=\omega$ by Gray in the case $n=2$ and Crans, Steiner, and Verity independently in the case $n=\omega$.

Edit: To clarify, these are all biclosed, and their right adjoints are the $n$-categories whose objects are strict $n$-functors and whose higher cells are (op)lax natural transformations and (op)lax modifications between them.

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  • $\begingroup$ I have edited my question to clarify what I mean by closed and by compact closed. $\endgroup$ Sep 1, 2018 at 18:20

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