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The inclusion $U$ of the (2,1)-category of symmetric monoidal categories with duals into the (2,1)-category of symmetric monoidal categories admits a left 2-adjoint functor $L$ for formal reasons.

In particular, for any symmetric monoidal category $C$, the unit map $C→L(C)$ exhibits $L(C)$ as the free symmetric monoidal category with duals generated by $C$.

Is there a citeable reference for any explicit construction of $L(C)$?

Generally speaking, we cannot expect the resulting functor $C→L(C)$ to be full, since the (formal) trace in $L(C)$ of a morphism in $C$ can give an endomorphism of the monoidal unit in $L(C)$ that does not come from $C$.

It seems to me that in the case when $C$ is a symmetric monoidal groupoid, it is reasonable to expect that the functor $C→L(C)$ is at least a faithful functor. Has this been shown explicitly?

What conditions on $C$ guarantee that $C→L(C)$ is a faithful functor?

I am particularly interested in the cases when $C$ is some symmetric monoidal category of infinite-dimensional vector spaces, such as Banach, Hilbert, or Fréchet topological vector spaces (possibly with invertible or unitary maps as morphisms), or (perhaps) solid modules over real numbers.

In the nonsymmetric case, the work of Delpeuch (Autonomization of monoidal categories) does give such an explicit construction of $L(C)$ and it proves that the inclusion functor $C→L(C)$ is a fully faithful strong monoidal functor, but this result does not seem to immediately extend to the symmetric case.

The forgetful functor $U$ factors through the (2,1)-category of traced symmetric monoidal categories, and the left adjoint functor from traced symmetric monoidal categories to symmetric monoidal categories with duals is given by the Int-construction of Joyal–Street–Verity (Traced monoidal categories, Section 4), who prove that the unit of this adjunction is a fully faithful strong monoidal functor.

However, I could not find any sources about free traced symmetric monoidal categories on symmetric monoidal categories, including conditions under which the unit map can be shown to be a faithful functor.

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    $\begingroup$ Can you explain why you expect it to be faithful for a groupoid ? $\endgroup$ Nov 7, 2021 at 12:08
  • $\begingroup$ @MaximeRamzi: In the free construction, invertible morphisms can be passed “through” units and counits, so it seems to me that given an arbitrary morphism in L(C) we can collect the morphisms of C in one place, composing them together. What remains is just a bunch of units and counits, which cancel each other. So if my arguments are correct, there is a very concrete explicit description of L(C): add units and counits for each object, as well as formal traces of all morphisms. This seemingly makes it easy to conclude that C→L(C) is faithful. $\endgroup$ Nov 7, 2021 at 15:56
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    $\begingroup$ Note that if you literally work with the (2,2)-category $SMC_{2,2}$ of symmetric monoidal categories, strong symmetric monoidal functors, and symmetric monoidal natural transformations, then the full sub-(2,2)-category $SMD$ of symmetric monoidal categories with duals for objects is in fact a (2,1)-category. So if $L_{2,2} : SMC_{2,2} \to SMD$ exists, then it carries every symmetric monoidal natural transformation to an invertible symmetric monoidal natural transformation. I'm convinced that $L_{2,1} : SMC_{2,1} \to SMD$ exists, but not so sure about $L_{2,2}$. $\endgroup$
    – Tim Campion
    Nov 11, 2021 at 11:44
  • $\begingroup$ @TimCampion: Coming from the (∞,1)-setting, I certainly had the (2,1)-category in mind. I edited the question accordingly. $\endgroup$ Nov 14, 2021 at 1:16

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In my thesis (which should be posted soon) I formally deduce the existence of "$L_{\infty,1}$", in an $(\infty,1)$-categorical setting; I do not provide an explicit construction. One point I exploit in my thesis is that there are several variants of $L_{\infty,1}$, respecting additional colimits and so forth, which also exist and are sometimes more what one wants.

I believe that $L_{2,1}C$ admits a description of the following form.

  • Objects: pairs $(a_+,a_-) \in Ob C \times Ob C$

  • Morphisms: $Hom((a_+,a_-), (b_+,b_-)) = \int^{x \in C} C(a_+ \otimes b_- \otimes x, b_+\otimes a_- \otimes x)$

(that's a coend in there.)

  • To compose $f : a_+ \otimes b_- \otimes x \to b_+ \otimes a_- \otimes x$ with $g : b_+ \otimes c_- \otimes y \to c_+ \otimes b_- \otimes y$, you just compose along the common $b_+$ part (after tensoring with appropriate identities and inserting a few swaps) to get a map $a_+ \otimes c_- \otimes (b_- \otimes x \otimes y) \to c_+ \otimes a_- \otimes (b_- \otimes x \otimes y)$

etc. The canonical functor $C \to L(C)$ sends $c$ to $(c,I)$ where $I$ is the unit, and the dual of $(c_+,c_-)$ is $(c_-,c_+)$.

If the above description is correct, perhaps you could find a faithfulness criterion using it.

I am having difficulty convincing myself that any explicit construction along the above lines actually has duals for objects, if it is well-defined at all.

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  • $\begingroup$ If C is a groupoid, this formula says that the set of morphisms from (a,I) to (b,I) can be computed as the disjoint union of C(a⊗x,b⊗x) over all isomorphism classes of objects x. In particular, the embedding of C(a,b) into this set is clearly injective. So it appears that the answer is positive in case of groupoids. $\endgroup$ Nov 14, 2021 at 0:52
  • $\begingroup$ If C already has duals, then formally adding duals should not change C (the operation of formally adding duals should be idempotent). Now, if C is a groupoid and C has duals then, according to Dmitri Pavlov's comment, the set of morphisms from (a,I) to (b,I) can be computed as the disjoint union of C(a⊗x,b⊗x) over all isomorphism classes of objects x. That's not C... Am I missing something? $\endgroup$ Nov 14, 2021 at 13:05
  • $\begingroup$ @AndréHenriques I haven't carefully considered Dmitri's comment (and to be honest, I have not carefully checked the above construction I outline), but note that if C is a symmetric monoidal groupoid with duals, then those duals must be inverses for all objects, and then maybe it's a bit more believable? EDIT: Hm... maybe my formula is wrong! $\endgroup$
    – Tim Campion
    Nov 14, 2021 at 14:18
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I thought the autonomization construction can be generalized to the symmetric case, and had a go at writing up the construction: https://arxiv.org/abs/2201.07527

However it does not work, because of an observation attributed to Plotkin: http://ncatlab.org/nlab/show/traced+monoidal+category#adding_traces

Because compact closed categories are also traced, this means we cannot hope for a faithful injection in general.

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    $\begingroup$ Based on your attempted proof, can we identify additional conditions that one has to impose on a symmetric monoidal category in order to get a faithful functor, e.g., so that your proof works? $\endgroup$ Jan 21 at 20:26
  • $\begingroup$ Not that I can immediately think of, but it is indeed an interesting question! $\endgroup$
    – pintoch
    Jan 22 at 15:54

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