# Universal example of Lie algebra

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 ?

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.

• 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 ? – d1592549 Nov 23 '19 at 16:27
• 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. – Adrien Nov 23 '19 at 16:43
• 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. – Adrien Nov 23 '19 at 16:46
• 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). – Denis Nardin Nov 24 '19 at 11:21
• @DenisNardin I didn't know that, thanks for pointing it out ! – Adrien Nov 24 '19 at 12:54