Rectifying the definition of a closed category

Question

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.

Answer 1

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.

Answer 2

While relevant references have been mentioned in the comments and in john's answer, I don't believe these fully answer the question as asked, at least without doing a little further reading, so here's a complete answer.

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In Example 3.2 of Day's 1974 An embedding theorem for closed categories, Day defines a closed category $(\mathcal C, [{-}, {-}], I)$ to be associative if the canonical natural transformation $$\hat L_{a, b, c, d} \colon \int^x \mathcal C(a, [x, d]) \times \mathcal C(b, [c, x]) \to \mathcal A(a, [b, [c, d]])$$ is invertible. This corresponds to asking for the induced promonoidal structure on $\mathcal C$ to be associative, as mentioned in john's answer. If $\mathcal C$ is a symmetric closed category, then it is automatically associative.

This observation was rediscovered in Uustalu–Veltri–Zeilberger's 2020 Eilenberg-Kelly Reloaded, where the authors give a correspondence between the stronger notion of closed category satisfying associativity, and monoidal categories (Theorems 2.10 and 3.8 ibid.). Their observation is motivated by Proposition 18 of Street's 2013 Skew-closed categories, which establishes a correspondence between skew-closed categories and skew-monoidal categories, as mentioned in john's answer.

Answer 3

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).