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Let $\mathcal{C}$ be a braided (not necessarily symmetric) monoidal category. Then we can define what monoids and commutative monoids in $\mathcal{C}$ are. What is the correct definition of a Lie algebra in $\mathcal{C}$? I saw many different ones, none of them focusing on the for me important aspect, which is Koszul duality (in the sense of GinzburgKapranov). So for $V \in \mathcal{C}$, I want $Comm(V) \cong T(V)/L(V)^{+}$ for the corresponding free objects (if they exist). I'm not an expert with operads, so I think I first need to understand how to define an operad in a braided monoidal category. I would be very thankful for some literature about that.

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    $\begingroup$ One keyword is "braided operads" which are to braided monoidal categories what operads and nonsymmetric operads are to symmetric monoidal and monoidal categories respectively. $\endgroup$
    – Phil Tosteson
    Dec 4, 2019 at 12:03
  • $\begingroup$ Can you really reasonably define commutative algebras in braided monoidal categories? $\endgroup$
    – Fernando Muro
    Dec 4, 2019 at 22:47
  • $\begingroup$ @PhilTosteson maybe I'm misunderstanding you but non-symmetric operads only make sense in symmetric monoidal categories, and similarly for braided operads. $\endgroup$
    – Fernando Muro
    Dec 4, 2019 at 22:48
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    $\begingroup$ @BipolarMinds No, even the definition of non-sym operad uses the commutativity constraint. $\endgroup$
    – Fernando Muro
    Dec 5, 2019 at 20:46
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    $\begingroup$ I think that most people trying to work with this kind of thing say that in the braided case one has a PROP, not an operad, see for example top of p.17 in arxiv.org/abs/hep-th/9303148 $\endgroup$
    – Vladimir Dotsenko
    Oct 16, 2020 at 5:48

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