3:35 PM
@AT0 Welcome! Just so you and others know, this room is mostly inactive these days; the conversation has migrated to a discord server, which can be accessed here: nodorek.net . I'd suggest that if you have further or follow-up questions you ask them there.
6
Regarding your question, I would say that the answer is yes. First of all, in general, operads in a monoidal (always implicitly ∞-)category V are meant to parametrize algebraic structures for objects of a V-enriched category. In particular, if we have a free/forget adjunction Spaces <==> V with monoidal left adjoint (such as when V is presentably monoidal) then an operad O in spaces yields an operad Free(O) in V.
However, in this case, for a V-enriched category C, Free(O)-algebras in C will be equivalent to O-algebras in the underlying unenriched category.
In your case, I am guessing that by "dgCat" you mean the (∞-)category of small pretriangulated dg-categories, but please correct me if I'm wrong. Assuming so, there is indeed an adjunction Spaces <==> dgCat, which is really a composite adjunction Spaces <==> Cat <==> dgCat. The first left adjoint considers a space as a category, and the second left adjoint takes a category A to its dg-category of "finite k-linear presheaves".
By this I mean the smallest pretriangulated dg-subcategory of Fun(A^{op},dgVect) containing the image of the "stabilized" Yoneda embedding, carrying an object a \in A to the dg-presheaf C_* hom_A(-,a).
(This is the Yoneda embedding postcomposed with the left adjoint C_* : Spaces --> dgVect.)
If A is in fact a space, this has an alternative description that you may appreciate. It suffices to assume that A is connected, since left adjoints commute with coproducts. Choose a basepoint of A. Then, Fun(A^{op},dgVect) is the ∞-category of dg-modules over the based loopspace, or equivalently over its chains, and the image of the "stabilized" Yoneda embedding is spanned by the regular representation.
