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The definition of a closed category I'm using is here.

Suppose $V$ is a closed category and that for each object $b\in V$, $[b,-]$ has a left adjoint $- \otimes b$. The result is nearly a monoidal category, but the associator $\alpha \colon (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c)$ is not generally an isomorphism.

Question: Can we modify the definition of closed category directly so that whenever the adjoint above exists, result is always closed monoidal?

I know that we can fix this by requiring the adjunction to be "internal" in the sense that we have a natural isomorphism $\Phi \colon [a \otimes b, c] \rightarrow [a, [b, c]]$ but I'd rather there not be an asymmetry between closed and monoidal categories, since we can get from monoidal to closed by only demanding an ordinary adjuntion.

Edit: In light of my answer below, I've decided this question needed clarifying. What I want is some additional structure or property added to the axioms of a closed category such that

1) If $V$ is monoidal and $- \otimes b$ has a right adjoint $[b, -]$, then $[-,-]$ makes $V$ into this modified closed category

2) If $V$ is this modified closed category and $[b, -]$ has a left adjoint $-\otimes b$, then $-\otimes-$ makes $V$ into a monoidal category.

In the current state of affairs, 1 holds but 2 does not. With the additional property proposed in my answer below, 2 holds, but 1 does not.

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  • $\begingroup$ Does this MO thread help? mathoverflow.net/questions/21382/… $\endgroup$ – David White Sep 22 '15 at 14:07
  • $\begingroup$ @DavidWhite, not quite. What I have here is the reverse question: I start with the internal hom and have a tensor that's adjoint to it. That question asks whether starting with the tensor and having an internal hom that's adjoint to it gives you a self-enriched category. $\endgroup$ – SCappella Sep 22 '15 at 20:15
  • $\begingroup$ I think this can help: (journals.cambridge.org/…) $\endgroup$ – Buschi Sergio Sep 22 '15 at 20:29
  • $\begingroup$ @BuschiSergio Am I right in saying that the answer I want is a promonoidal category such that associativity is an isomorphism and that the product P:V^op x V^op x V -> Set is representable, i.e. P(a,b,c) = V(a,[b,c]) for some [-,-]: V^op x V -> Set? If that's so, you can write it up as an answer and I'll accept it. $\endgroup$ – SCappella Sep 23 '15 at 5:58
  • $\begingroup$ I wish only say that for a symmetric closed category $V$ you have a immersion on a symmetric monoidal closed one $W$. This immersion "seem" dense (as I read from article), then the restriction to $V$ of monoidal product of $W$ is just the your (defined as left adjoint..). Then is easy to get a isomorphism $[a\otimes b, c]\cong [a, [b, c]]$ on $V$. $\endgroup$ – Buschi Sergio Sep 23 '15 at 14:08
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A symmetric closed category is a closed category together with isomorphisms $$s:[A,[B,C]] \cong [B,[A,C]]$$ satisfying a few axioms: see Definition 1.1 of the paper ``On embedding closed categories" by Day and Laplaza that Buschi Sergio mentioned.

If you have one of these $(C,[-,-],I)$, together with adjoints $- \otimes A \dashv [A,-]$, then you get a symmetric monoidal category $(C,\otimes,I)$. It is possible to give an elementary argument proving this, but it takes a bit of work to write down.

It seems that you have realised this in your discussion with Buschi Sergio already but let me point it out anyway. The result follows from Proposition 2.2 of Day and Laplaza's paper which asserts that a symmetric closed category gives rise to a symmetric promonoidal one with promonoidal structure $$P(A,B;C)=C(A,[B,C])$$: that is, a symmetric pseudomonoid in the symmetric monoidal bicategory Prof of profunctors. Now if you have the adjoints you then have a representable symmetric promonoidal structure $$C(A \otimes B,C) \cong P(A,B;C)$$ and such amounts to a symmetric monoidal structure on $C$. (Because the strong symmetric monoidal pseudofunctor Cat → Prof is essentially fully faithful - i.e. locally an equivalence).

The same paper gives examples showing that the canonical associator for the (almost) monoidal structure associated to a closed category needn't be invertible. So there is a genuine asymmetry between the definitions of monoidal and closed category.

The notions of skew monoidal and skew closed category rectify the asymmetry in a different way -- there is a perfect correspondence between skew monoidal structures $(C,\otimes,I)$ and skew closed structures $(C,[-,-],I)$ related by a natural isomorphism $C(A \otimes B,C) \cong C(A,[B,C])$. See Proposition 18 of the paper "Skew closed categories" http://arxiv.org/abs/1205.6522 by Ross Street.

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  • $\begingroup$ I think I understand now, but a point of clarification, in my question, I'm not talking about symmetric monoidal categories, only monoidal categories. What happens when you use a non-symmetric promonoidal category and demand that it's representable? Do non-symmetric promonoidal categories exist? $\endgroup$ – SCappella Sep 29 '15 at 19:30
  • $\begingroup$ (1)Then you get a non-symmetric monoidal category. (2) Yes, non-symmetric promonoidal categories exist - if C is a non-symmetric monoidal category then setting $P(A,B,C)=C(A \otimes B,C)$ gives a non-symmetric promonoidal category. $\endgroup$ – john Sep 30 '15 at 14:46
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This isn't quite what I want for reasons I'll explain below but...

Suppose we require that the functor $H = \mathrm{Hom}(I, -) \colon V \rightarrow Set$ is left-cancellable in the sense that for functors $F, G \colon C \rightarrow V$, if $H \circ F$ is naturally isomorphic to $H \circ G$, then $F$ is naturally isomorphic to $G$. Note that this can be completely expressed in a plain old closed category.

Then, when $[-,b]$ has a left adjoint $-\otimes b$, we have $\mathrm{Hom}(I, [a \otimes b, c]) \simeq \mathrm{Hom}(a \otimes b, c) \simeq \mathrm{Hom}(a, [b,c]) \simeq \mathrm{Hom}(I, [a, [b,c]])$. Under our hypothesis, $[a \otimes b, c] \simeq [a, [b, c]]$, which as I noted in the question, implies that $\otimes$ makes $V$ into a monoidal category.

However, I don't think I'll count this because there are lots of closed monoidal categories where the hypothesis doesn't hold (an example is $\mathbb{R}$ considered as a poset with addition as the monoidal product and subtraction as the internal hom).

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