# Definition of $E_{n}$-operad in dgCat

In "Derived Algebraic Geometry and Deformation Quantization" Toën defines in 5.1.2 an $$E_{n}$$-monoidal A-linear dg-category as an $$E_{n}$$-monoid in the symmetric monoidal $$\infty$$-category $$dgCat_{A}$$ of compactly generated (A-linear) dg-categories.

Concretely, unwrapping this definition Toën says this is equivalent of having a dg-category $$T\in dgCat_{A}$$ and morphisms $$E_{n}(k)\otimes T^{\otimes k}\to T$$ satisfying the usual conditions of an algebra over an operad.

Question: What are these dg-categories $$E_{n}(k)$$?

I thought about turning the $$\mathbb{E}_{n}$$ operad defined by Lurie in Higher Algebra into a dg-category, but Im unsure if this would be correct or if it would be relatively easier than giving a direct definition.

I'm not very experienced in operads in general and $$\infty-$$operads in particular, so I apologize if the question has an immediate answer or if it comes from a fundamental misunderstanding of the topic.

• In Toën's paper, E_n(k) is not a dg-category, but merely a space (he says so explicitly in 5.1.2). This space is just the kth space of the operad E_n. Aug 19, 2021 at 5:08
• I bet it's an E_n-algebra in the Chain-enriched category of graphs over a fixed object set. I mean the chain E_n operad. Aug 19, 2021 at 9:58
• @DmitriPavlov Thank you. I think the source of my confusion comes from him saying the tensor by E_n(k) is taken inside dgCat. This is where my ignorance is showing, but I interpreted this as either E_n(k) being a dg-category, or then considering dgCat as a symmetric monoidal category tensored over the symmetric monoidal category of spaces, I interpret your answer as the latter, but then how is this action given?
– AT0
Aug 19, 2021 at 10:15
• @AT0: Any presentable ∞-category has a canonical tensoring over spaces: given a simplicial set S and an object A, the tensoring S⊗A can be defined as hocolim_{s∈Δ/S} A, for example. Aug 19, 2021 at 12:44
• Adding on @DmitriPavlov answer, in this specific case of an $A$-linear presentable category $C$ and a space $U$, the infinity category $C \otimes U$ is just $Fun(U,C)$, the $\infty$-category of functors from the $\infty$-groupoid $U$ to the $\infty$-category $C$. In other words the colimit and limit over a space here are the same, and these are just "$C$-valued local systems over $U$". Aug 21, 2021 at 21:09