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See [LV] Loday–Vallette, Algebraic Operads, Section 8.6 for all the definitions. … Let $\mathtt{P} = \mathbb{F}(V)/(R)$ and $\mathtt{Q} = \mathbb{F}(W)/(S)$ be two quadratic operads presented by generators and relations, and let $$\lambda : W \circ_{(1)} V \to V \circ_{(1)} W$$ be a …

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 …

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 …

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. …

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 …

on the arXiv last week (1807.11671) proving that $E_2$ is not formal over $\mathbb{F}_2$ as a non-symmetric operad (what he calls planar operad), i.e. you cannot find a zigzag of quasi-isomorphisms of operads …

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 …

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. …

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( …

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

An object $X \in \mathsf{C}$ has two operads "naturally" (the two constructions aren't functorial) associated to it: the operad of endomorphisms and the operad of coendomorphisms $$\mathtt{End}_X(r) = … And what about topological operads, say, where things cannot be dualized as easily? (Asked in February on math.SE) …

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

"Moduli spaces and formal operads" in Duke Math. J. 129, No. 2, 291–335 (2005) …

Wahl, Framed discs operads and Batalin-Vilkovisky algebras. Q. J. Math., 2003, 54, 213-231"). These two operads are not weakly equivalent, and their categories of algebras are different. … More generally, $\mathtt{D}_n(1)$ is contractible, whereas $\mathtt{fD}_n(1) \simeq \mathrm{SO}(n)$ is non-contractible, so the operads cannot be weakly equivalent. …

The recognition principle [Boardmann–Vogt, May] states that a grouplike algebra over the little $n$-disks/cube operad is weakly equivalent to an $n$-fold loop space. There are technical hypotheses tho …