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In the recent IAS talk (available here: https://www.youtube.com/watch?v=LeaiPHAh0X0 - from 45:20) Lurie mentioned an (additive monoidal, with all colimits) category $\mathcal E$ together with a Lie algebra object $L_U$ in it such that for any other (additive monoidal, with all colimits) category $\mathcal C$ there is an identification between Lie algebra objects in $\mathcal C$ and functors $F{:}\mathcal E\to\mathcal C$ that preserve monoidal structure and colimits. It looks similar to a classical story of the "walking monoid" $\Delta$ and monoid objects in monoidal categories. I'm interested in some references on this category $\mathcal E$, is it a well known thing ?

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For any operad $O$ there is a symmetric monoidal category $P(O)$ constructed as follow:

  • the set of objects is $\mathbb{N}$
  • the tensor product is given by addition and the symmetry by the equality $m+n=n+m$
  • then there is a unique way to define morphisms in such a way that $$Hom(n,1)=O(n).$$

You can look at https://arxiv.org/abs/math/0005197 for a more concrete construction. This is nothing but the free (symmetric monoidal category with an $O$-algebra), or equivalently the free PROP on $O$.

Then by construction $1\in P(O)$ is an $O$-algebra and if $S$ is any symmetric monoidal category, an $O$-algebra in $S$ is the same as a symmetric monoidal functor $P(O)\rightarrow S$. In a way this should be thought as a nice way of defining the notion of $O$ algebra in arbitrary symmetric monoidal categories. Formally $P$ is the left adjoint of the forgetful functor from PROPs to operads.

Now if $O(n)$ is an abelian group or a vector space you can take the additive/idempotent completion of $P(O)$, then formally add all colimits if you like, or replace everything by $\infty$-stuffs etc.. and you'll get the free symmetric monoidal (adjective) category on $O$ where adjective is whatever world you choose to work in.

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  • $\begingroup$ Thanks! This is fairly general and straightforward. So you mean that for $\mathcal E$ we can just take $P(L)$ (plus all of extra structure as you mentioned), with $L$ being Lie operad ? $\endgroup$
    – d1592549
    Nov 23 '19 at 16:27
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    $\begingroup$ Right, of course I should have said that, if $O=Lie$ you get what you were looking for, and this case is actually worked out in details in the paper I linked to. If $O=Ass$ you get the walking monoid. $\endgroup$
    – Adrien
    Nov 23 '19 at 16:43
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    $\begingroup$ It might also be useful to think of this as the analog of regarding an algebra $A$ as a category with one object. That way every algebra is endomorphism of something. In the same way this shows that every operad is the endomorphism operad of something. $\endgroup$
    – Adrien
    Nov 23 '19 at 16:46
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    $\begingroup$ In Lurie's works, this construction is usually referred to as the "symmetric monoidal envelope" of the operad (just adding it because it might be useful as a search term). $\endgroup$ Nov 24 '19 at 11:21
  • $\begingroup$ @DenisNardin I didn't know that, thanks for pointing it out ! $\endgroup$
    – Adrien
    Nov 24 '19 at 12:54

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