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It's often very convenient for objects in a monoidal category to have duals. Hence, it's natural to wonder whether an arbitrary monoidal category can be embedded in one where all objects have duals. For the usual notion of dual in symmetric monoidal categories, a complete answer to this was given by Joyal, Street, and Verity: any full monoidal subcategory of a compact closed monoidal category (that is, a symmetric monoidal category in which all objects have duals) is a traced monoidal category, so being traced is a necessary condition for embeddability in a compact closed category — and it is also sufficient, by the "Int-construction".

Now suppose we relax the notion of dual, replacing compact closed categories by *-autonomous ones (but keeping the symmetry, for simplicity). Every object of a *-autonomous category still has a "dual" in a useful sense, but since the unit and counit of the duality use different monoidal structures, no "traces" are induced. And indeed, in this case, every closed symmetric monoidal category with pullbacks can be embedded in a *-autonomous category, namely its Chu construction relative to an arbitrary choice of dualizing object. (The condition "with pullbacks" can be replaced by "with finite products" if we take the dualizing object to be the terminal one.)

So far, so good. But a significant part of the value of having dual objects is that they can be used to construct the internal-homs as $[A,B] = (A \otimes B^*)^*$. So it's natural to ask whether an arbitrary closed symmetric monoidal category can be embedded in one with duals in a way that preserves the closed structure. In the compact closed case, we're out of luck: since $A^* = [A,I]$ in a compact closed category, any sub-closed-monoidal-category of a compact closed category is itself already compact closed.

But in the *-autonomous case there's room for a nontrivial answer. For instance, I believe that if $\bot$ is an object of a closed symmetric monoidal category $\mathcal{C}$ such that $[[A,B],\bot] \cong A\otimes [B,\bot]$ for all $A,B$, then the embedding of $\mathcal{C}$ in ${\rm Chu}(\mathcal{C},\bot)$ preserves the internal-homs. This is a strong condition, of course, but it's satisfied for instance if $\bot$ is a zero object. Thus, any closed symmetric monoidal category with pullbacks and a zero object can be closedly-embedded in a *-autonomous one.

On the other hand, this sufficient condition is certainly not necessary, since not every *-autonomous category has a zero object. Indeed, I don't know of any nontrivial necessary condition. That is, I don't know any condition that can be expressed in the language of closed symmetric monoidal categories that is true of full closed-monoidal-subcategories of *-autonomous categories but is not true of all closed symmetric monoidal categories. Hence my question:

What is a necessary and sufficient condition on a closed symmetric monoidal category (feel free to assume it has any limits and colimits you like) for it to be closed-monoidally-embeddable in a *-autonomous category?

Failing that, is there any nontrivial necessary condition for such an embedding to exist?

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  • $\begingroup$ It seems pretty strong to me to ask for internal homs to be preserved -- it doesn't quite imply the embedding is fully faithful, but it feels "close" to this. I doubt, for instance, that $(FinSet,\times)$ embeds into a $\ast$-autonomous category in this sense. $\endgroup$ – Tim Campion Oct 7 '19 at 12:02
  • $\begingroup$ Also, if $\bot$ is a zero object, then isn't $Chu(C,\bot)$ just $C \times C^{op}$? $\endgroup$ – Tim Campion Oct 7 '19 at 12:12
  • $\begingroup$ @TimCampion Yes, if $\bot$ is terminal, then ${\rm Chu}(C,\bot) = C\times C^{\rm op}$. I agree, it does seem like a strong condition... but a priori I wouldn't have expected $(\rm Ab,\otimes)$ to embed closedly in a $\ast$-autonomous category either, and yet apparently it does (unless I made a mistake). So the question is, can you isolate a property of $({\rm FinSet},\times)$ that prevents it from admitting such an embedding? $\endgroup$ – Mike Shulman Oct 7 '19 at 22:43
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In my preprint *-autonomous envelopes I showed that any closed symmetric monoidal category is closed-monoidally-embeddable into a $\ast$-autonomous category, using a variation of Hyland's polycategorical envelope. Thus, there is no additional necessary condition.

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