# 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 ?

## 1 Answer

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 ? 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. 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. 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). Nov 24 '19 at 11:21
• @DenisNardin I didn't know that, thanks for pointing it out ! Nov 24 '19 at 12:54