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The comment of Tom Goodwillie is correct. In the book, a connected weight graded operad is one that decomposes as: $$\mathcal{P} = \mathbb{K} \mathrm{id} \oplus \mathcal{P}^{(1)} \oplus \mathcal{P}^{( …

I'll use the standard notation that $\mathrm{Pois}_n$ is the usual $n$-Poisson operad. I'll assume that you mean $(\mathrm{Pois}_n)_\infty = \Omega(\mathrm{Pois}_n^¡)$. Then no, $(\mathrm{Pois}_n)_\in …

Fresse, Homotopy of Operads and Grothendieck–Teichmüller Groups, Mathematical Surveys and Monographs 217. https://bookstore.ams.org/surv-217/ However, the idea that topological spaces are obsolete and … Or the little disks/cubes operads? Simplicial sets of course have their place, but there are times when you just can't avoid topological spaces. …

The proofs I know for $n \ge 3$ all involve operads. More precisely, the little disks operads, whose components are homotopy equivalent to the configuration spaces above. … The theorem is that these operads are formal. Hence the configuration spaces are formal. …

Yes, sure. You can take the free $\mathcal{O}$-algebra on your set of generators, and mod out by the $\mathcal{O}$-ideal generated by your relations. Then you get an algebra $A$ presented by generato …

Let me summarize the comments. You have several possibilities: [Ryan's comment] You can consider the sub-operad $fD'_n \subset fD_n$ such that $fD'_n(r) = fD_n(r)$ for $r \ge 2$, and $fD'_n(1) = SO( …

But I think you can use a strategy similar to what is done in the paper The Intrinsic Formality of $E_n$-operads by Fresse and Willwacher. I'm not saying it's easy, but it's systematic. …

It appears when defining semi-direct products of operads (as in the paper of Salvatore and Wahl), for example. …

Let me mention that this is related to this earlier question of mine (which is unanswered :-( ) and more generally to semi-direct products of operads by bialgebras. …

As written in the paper that you cite, this construction originates from two papers of McClure–Smith, and a geometric interpretation of the table reduction morphism is given in: Clemens Berger and …

There is also Horel's paper Operads, Modules and Topological Field Theories. …

This is the homotopy version of the following data: a collection of spaces $\{X_c\}_c$ for all colors $c$; units $e_c \in X_c$; multiplications $- \cdot_c - : X_d \times X_{d'} \to X_c$; satisfyin …

Inside an aerial vertex, you can insert a unicolored graph, of the kind found in Kontsevich's paper Operads and motives in deformation quantization. …

Since operads have units, this is equivalent to giving composition maps $\circ_i : M(k) \otimes P(l) \to P(k+l-1)$ for $1 \le i \le k$ (simply put, $m \circ_i p = m(1,\dots,1,p,1,\dots,1)$ where $1$ is …

They give several examples, and the paper is a great motivation for why one would be interested in getting operads in schemes (because under good circumstances, they are formal). …