Universal example of Lie algebra

Question

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 ?

Answer 1

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.