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the forgetful functor from nonsymmetric cyclic operads to nonsymmetric operads, and $B^{cyc}$ resp. … Everything above is true mutatis mutandis for cyclic symmetric operads vs usual symmetric operads. …

At the risk of saying something stupid I'm promoting my comment to an answer. In terms of the Stasheff polytopes the $A_n$ operad sits inside the $A_\infty$ operad as the union of all faces of dimensi …

(In particular the phenomenon is not something general for any sequence of Koszul operads - you really need a PBW theorem in the background.) Here is how it goes. …

Also, a good jumping-in point could be Ginzburg and Kapranov's "Koszul duality for operads". …

There is no possibility of encoding the existence of the inverse in the language of operads. … More generally, operads and their relatives are all about encoding operations and relations which are multilinear, and inversion is just the simplest example of something nonlinear. …

Do you know Theorem 1.4 ("relative formality") of Lambrechts-Volic paper on formality of the small disks? It says that when $m \geq 2n+1$ the $\infty$-morphism is in fact a morphism and it is the obvi …

In brief, already Cisinski and Moerdijk ("Dendroidal sets and simplicial operads", arXiv:1109.1004) proved a Quillen equivalence between simplicial operads and dendroidal sets. … equivalent to Lurie's ∞-operads. …

Yes, absolutely. I really think the best place for learning this is the book by Loday and Vallette. This example fits more generally into the following context. Let $P$ be a Koszul operad and $A$ a $ …

We worked out some answers to this question in our paper: arXiv:1904.03585 (Edit: the following answer refers to v1 of the paper on the arXiv!) Here's the short version. There are two possible natura …

When $n$ is even these spaces are complex algebraic varieties, so the cohomology comes with a mixed Hodge structure. Moreover, this mixed Hodge structure is pure: the cohomology ring is generated in d …

I also noticed this at some point. I think you are right. One reference where this assumption is explicitly spelled out is the paper of Getzler and Jones.

In fact the tensor product of two $A_\infty$ algebras can be made into an $A_\infty$ algebra in an explicit way: there are two constructions, one by Saneblidze-Umble and one by Loday. See the paper ht …

The original reference is Marta C. Bunge, "Coherent extensions and relational algebras", Trans. Amer. Math. Soc. 197 (1974), pp. 355-390.

It is a typo. The map $f$ should only be assumed to be a morphism of the underlying graded $\mathbb S$-modules.