The Chu-construction and the Int-construction

The Chu construction is a way of building a star-autonomous category $\mathrm{Chu}(C,\bot)$ from any closed monoidal category $C$ with pullbacks and a choice of an object $\bot\in C$ to become the dualizing object. If $\bot$ is the terminal object, then the underlying category of the Chu construction is just $C\times C^{\mathrm{op}}$.

In their paper Traced monoidal categories, Joyal, Street, and Verity described a construction of a compact closed category $\mathrm{Int}(C)$ from any traced monoidal category $C$. The objects of $\mathrm{Int}(C)$ are pairs $(X,U)$ of objects of $C$, and its morphisms $(X,U) \to (Y,V)$ are morphisms $X\otimes V \to Y\otimes U$ in $C$, with composition defined using the trace.

There seems to me to be some kind of family resemblance between these constructions. In both cases we are "formally" or "freely" adding some kind of "duals" to a category $C$ by constructing a category whose objects are pairs of objects of $C$, one treated covariantly and one contravariantly. However, the differences are substantial, and I haven't been able to make this similarity precise in any way.

Is there any formal relationship between the Chu construction and the Int construction?

It seems to me that, if $C$ is a traced monoidal category (with unit $I$) that is also closed, there is a comparison functor $Chu(C,I)\to Int(C)$, which is perhaps what you had in mind. However, this functor does not seem particularly well-behaved: in general, it is neither full nor faithful, and only lax monoidal. (It also isn't essentially surjective on objects, for whatever that's worth.) [EDIT: You're right, it isn't even lax monoidal in general, but it might be oplax monoidal.]

Consider the discrete case: if (the underlying category of) a traced monoidal category $C$ is discrete, then it is just a cancellative commutative monoid, and $Int(C)$ is the free abelian group on it; however, in this case, $Chu(C,1)$ (where $1$ denotes the unit of the monoid) is the cofree abelian group on it.

In fact, a quarter-century ago, Dusko Pavlovic proved that the Chu construction is always (in a suitable sense) the cofree way of turning a pointed symmetric monoidal closed category into a symmetric $*$-autonomous one. Which stands in stark contrast to $Int$ being the free way of turning a traced monoidal category into a compact closed category.

One way this shows up is that there is always a forgetful functor $Chu(C,\bot)\to C$---which is the counit of the adjunction---but no obvious embedding $C \to Chu(C,\bot)$, whereas there is always an embedding $C \to Int(C)$---the unit of the adjunction---but no obvious forgetful map $Int(C)\to C$. In conclusion, I don't think that $Int$ and $Chu$ are really all that similar after all.

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Since you ask, here are some more details.

As you said in your question, objects of $Chu(C,\bot)$ and $Int(C)$ comprise two $C$-objects: one treated covariantly (the "positive space"), the other, contravariantly (the "negative space"); moreover, the duality in each case is given by swapping the two. So it seemed to me that if you want to highlight these similarities, one should build a comparison functor that preserves these very features.

It turns out---again, you were right about this---that there's no real reason to restrict to the case $\bot=I$; I did so only because I was hoping to build a lax monoidal functor, which would require at least a map $\bot\to I$, and was too lazy to check if the usual coherence conditions on such a map were enough to make this work. But since I've lost hope of making the comparison functor lax monoidal, there's no point for this restriction at all. (To make the functor oplax monoidal would require a suitably coherent map $I \to \bot$ instead.)

Anyways, the naïve thing to way to build a functor $Chu(C,\bot)\to Int(C)$ would be as a kind of forgetful functor, where you forget the "inner product" of a Chu space, and remember only the positive space and the negative space. Now a map of Chu spaces comprises two $C$-arrows: one between the positive spaces, and one between the negative spaces; tensoring these two maps together produces a $C$-arrow of the right type to define a map in $Int(C)$. So one can ask if this defines a functor. Amazingly, it does in the symmetric case; moreover, one simply needs to add a $-2\pi$ twist to the negative component to make it work in the balanced case.

I'll append a photograph of my calculation, but first, let's settle notation. A Chu space $A$ comprises a positive space $a_+$, a negative space $a_-$, and an inner product $\alpha:a_-\otimes a_+\to\bot$; let $F(A)=(a_+,a_-)$ denote the corresponding object of $Int(C)$. A morphism of Chu spaces $\omega: A=(a_+,a_-,\alpha)\to B=(b_+,b_-,\beta)$ comprises a map $\omega_+:a_+\to b_+$ and a map $\omega_-:b_-\to a_-$ satisfying a diagram which is irrelevant to the purpose at hand; $\omega_+\otimes(\omega_-\circ\xi_{b_-}^{-1}):a_+\otimes b_- \to b_+\otimes a_-$ can be construed as a map $F(\omega):F(A)\to F(B)$ in $Int(C)$, where $\xi$ denotes the balance.

I claim that $F$ preserves identities and composition. Identities are easy: the identity Chu morphism is given by a pair of identities, and the identity map $(a_+,a_-)\to(a_+,a_-)$ in $Int(C)$ is given (not by the identity on $a_+\otimes a_-$, but) $\mathrm{id}_{a_+}\otimes\xi_{a_-}^{-1}$ ["Traced monoidal categories", section 4]. As for composition, see attached:

• Would you be able to write out the definition of your comparison functor? I had thought of such a functor in that direction, but it wasn't clear to me how to relate composition in the two cases --- I guess that is why you're restricting to the case where the dualizing object is the unit? And it wasn't clear to me that the tensor products were at all related; lax monoidal is more than I had hoped for. Oct 25, 2017 at 22:25
• Is there any way to derive that comparison functor from the universal properties of Chu and Int as a right and left adjoint respectively? I suppose it seems unlikely since the variance is wrong: we're mapping out of a right adjoint into a left adjoint. Oct 25, 2017 at 22:25
• If $U$ has both a left adjoint $F$ and a right adjoint $G$, then it can happen that the composite of the counit $UG\to\mathrm{Id}$ with the unit $\mathrm{Id}\to UF$ is "in the range of $U$", in the sense that it is $U\lambda$ for some natural transformation $\lambda:G\to F$. (For instance, where $U$ is the forgetful functor from groups to monoids, we get an embedding of the cofree group on M into the free group on M, which is a homomorphism.) But that's not what's happening here as the composite $Chu(C,\bot)\to C\to Int(C)$ maps $(a_+,a_-,\alpha)$ to $(a_+,I)$, not $(a_+,a_-)$. Oct 26, 2017 at 18:17
• Actually, let $P$ be the composite $Chu(C,\bot)\to C\to Int(C)$ described above, and let $N$ be its de Morgan dual $N(A):=(P(A^*))^*$. Then the $F$ that I constructed above is $F(A)=P(A)\otimes N(A)$---i.e., the natural composite $Chu(C,\bot) \to Chu(C,\bot) \times Chu(C,\bot) \to Int(C) \times Int(C) \to Int(C)$, where the middle arrow is $P \times N$. I don't think there's anything deep in that, but it makes the string diagram calcuation superfluous. Oct 26, 2017 at 18:27
• UPDATE: I think I've proven that, if $\bot$ is a monoid in $C$, then one can define a rival monoidal structure $\boxtimes$ on $Chu(C,\bot)$ such that $F(A \boxtimes B)=F(A)\otimes F(B)$. (There are several finicky details to be checked.) I strongly doubt that $\boxtimes$ is closed. In the case $\bot=I$ (and perhaps slightly more generally), there seems to be a coherent natural transformation $\otimes\to\boxtimes$, in which case we get the desired oplax monoidal structure $F(A \otimes B)\to F(A \boxtimes B)=F(A)\otimes F(B)$. I will let you know when I can: I'm quite excited about it! Oct 26, 2017 at 21:03