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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.

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    $\begingroup$ 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. $\endgroup$ Aug 19, 2021 at 5:08
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    $\begingroup$ 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. $\endgroup$ Aug 19, 2021 at 9:58
  • $\begingroup$ @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? $\endgroup$
    – AT0
    Aug 19, 2021 at 10:15
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    $\begingroup$ @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. $\endgroup$ Aug 19, 2021 at 12:44
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    $\begingroup$ 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 $U$". $\endgroup$
    – S. carmeli
    Aug 21, 2021 at 21:09

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