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Suppose we work with symmetric sequences over a suitable monoidal category. Given quadratic data $(E,R)$ PBW and Gröbner bases are of great practical importance to understand the quadratic operad $\m …

This is a short question: In the symmetric monoidal category of chain complexes (over a field if necessary), does the endomorphism operad carries a Hopf structure, i.o.w. can it be considered as a H …

This however is not true, since both $\mathcal{A}ss$ as well as $\mathcal{C}om$ are finite dimensional in each arity and there is no morphism of operads $\mathcal{C}om\to\mathcal{A}ss$. …

The theory of symmetric operads in chain complexes (say over a good enough field) is in some sense nice, because we have a well defined homotopy theory. … In particular we have a notion of infinity-morphisms of operads (maybe called homotopy morphisms instead), which can be defined as a cooperad map between the appropriate bar constructions of the operads …

If necessary, we can restrict the following to the case where we consider only Hopf (co)operads in the category of chain complexes over fields of characteristic zero. … In case of ordinary operads, their algebras can be defined as operad maps. …

I'm fishing in troubled waters here and therefore the question is vague and meant to be as general as possible. In particular "$e_n$-like operad" can be an algebraic or topological $e_n$ operad, as fo …

Moreover we have a model structure for operads, we can speak about homotopy operads, infinity morphism of operads ect. In such a setting. … Is there any interesting structure on the set of all possible (homotopy) $E_n$-operads? I know, this question is far from being precise. …

2$, let $\mathcal{E}_{n}$ be the $E_{n}$-suboperad of the Barratt-Eccles operad $\mathcal{E}$, $\mathcal{E}_{n}^{i}$ its Koszul dual cooperad in the sense described in the paper "Koszul duality of En-operads … Proof: Since $\mathcal{E}_{n}$ is an $E_{n}$-operad, by the definition of $E_{n}$-operads there is a zig-zag of quasi-isomorphisms of dg-operads $ \mathcal{E}_{n}\overset{\simeq}{\longleftarrow}\bullet …

Suppose we have two dg symmetric Koszul operads, say $O_1$ and $O_2$. …

This is a short question: Is it just unproven folklore (yet), or is it definitively known that $E_n$-operads are not formal, if the characteristic of the underlying field is not equal to zero? …

As a non expert in the theory of topological operads, I find it pretty hard, to understand what algebras for little balls/cubes/something operads are. … operads. …

In Loday and Vallette's book on algebraic operads, they use the "Endomorphism cooperad $End^c_{s\mathbb{K}}$", where $s\mathbb{K}$ is the base field, shifted into (homological) degree one. …