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$\newcommand{\p}{\mathcal{P}}$Let G be a group, and let $G_n$ denote the wreath product $G^n \rtimes \Sigma_n$. There seems to be a notion of a PROP $\p$ where the role of the symmetric groups $\Sig …

Let $B$ be the symmetric monoidal category of finite sets and bijections with disjoint union. Let $C$ be a symmetric monoidal category. Is there a standard name for a lax monoidal functor $F:B \to C$? …

Merkulov, Markl and Shadrin (Wheeled PROPs, graph complexes and the master equation) give a definition for wheeled properads, which are basically modular operads with oriented edges, which are constrained …

But what really got me scratching my head was the following statement from page 2 of Getzler & Kapranov's paper on modular operads (sorry for the lengthy quote): [...] …

Let $A$ and $B$ be $A_\infty$-algebras. It's true, but it's a quite nontrivial fact, that the tensor product $A \otimes B$ can be given the structure of $A_\infty$-algebra, too. What is much easier to …

Let $A$ be a dg algebra, say over a field. The Homotopy Transfer Theorem says that $H(A)$ can noncanonically be given the structure of $A_\infty$-algebra, extending the induced multiplication on $H(A) …

The connection to the little disk operad is that one can quite easily write down a morphism of operads $$ \PaB \to \Pi_1(D_2),$$ where $\Pi_1(-)$ denotes the fundamental groupoid, such that $\PaB(n) \ …

I have a particular kind of algebraic structure that's come up in my work. It's basically a chain complex equipped with a multiplication which is commutative and associative up to homotopy in a partic …