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I'm skimming through Turaev's "Quantum invariants of knots and 3-manifolds". One of the main results is Theorem 2.5. In my opinion, this Theorem is conceptually half-baked: 1) The ribbon structure on $\mathrm{Rib}_\mathcal{V}$ is not used. 2) We don't need a ribbon structure on $\mathcal{V}$ in order to describe functors on $\mathrm{Rib}_\mathcal{V}$. Therefore, I propose the following generalization:

Theorem(?). Let $\mathcal{V}$ be a small strict monoidal category with duality. Then the category of $\mathcal{V}$-colored ribbon graphs $\mathrm{Rib}_\mathcal{V}$ is a strict ribbon category and $i : \mathcal{V} \to \mathrm{Rib}_\mathcal{V}$, $V \mapsto (V,+)$, $f \mapsto$ (a coupon colored by $f$) is a morphism of strict monoidal categories with duality. It is universal: If $\mathcal{C}$ is any strict ribbon category and $F : \mathcal{V} \to \mathcal{C}$ is a morphism of strict monoidal categories with duality, then there is a unique morphism of strict ribbon categories $\overline{F} : \mathrm{Rib}_\mathcal{V} \to \mathcal{C}$ with $\overline{F} \circ i = F$.

Is this correct? If yes, does someone know a reference for this result?

Sketch of proof. On objects we map for example $((V_1,+),(V_2,-),(V_3,+))$ to $F(V_1) \otimes F(V_2)^* \otimes F(V_3)$. As for morphisms, we decompose ribbon graphs into "basic" ribbon graphs $[f]$, $\cup$, $\cap$, $\varphi$ with appropriate colorings $V_i$ and map them to the morphisms $f$, $b$, $d$, $\theta$ w.r.t. the objects $F(V_i)$. The hard part is to show that every isotopy between colored ribbon graphs is a finite sequence of certain moves which correspond to commutative diagrams in $\mathcal{C}$.

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  • $\begingroup$ Reading this, I have an addition question. Rib is a functor on the category of strict monoidal categories with duals. It is also a monad. What are the properties of this monad, and how much information does it encode about ribbon structure? $\endgroup$
    – Dimitri Chikhladze
    Commented Jun 29, 2014 at 12:38

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EDIT: I misunderstood what the OP was suggesting. This is probably true. I doubt anyone has bothered to write it, though I could be wrong. I suspect it's "well-known." The isotopies should just come from the Reidemeister moves, and sliding coupons across strands, which all are indeed known to be commutative diagrams in all ribbon categories.

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  • $\begingroup$ Hm? I use the definition of a duality as in Turaev's book, this contains $b$ and $d$. And I don't see why your observation gives a counterexample to my Theorem(?). I don't claim that $i$ is fully faithful! I think the whole idea of the construction of $\mathrm{Rib}_\mathcal{V}$ from $\mathcal{V}$ seems to be that we adjoin braidings (corresponding to tangles) and twists (corresponding to framings). $\endgroup$
    – Martin Brandenburg
    Commented Jun 28, 2014 at 22:33
  • $\begingroup$ +1 to the edited version. $\endgroup$
    – Theo Johnson-Freyd
    Commented Jun 29, 2014 at 4:30

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