The paper concerns the Gelfand-Kirillov dimension and generating series of nonsymmetric operads. An analogue Bergman's gap theorem is proved, namely, no finitely generated locally finite operad has strictly between $1$ $2$. For every $r\in \{0\}\cup \{1\}\cup [2,\infty)$ or $r=\infty$, we construct a single-element with $r$. We also provide counterexamples to two expectations Khoroshkin Piontko...

Abstract Recall that a dendroidal set X is an ∞-operad if it satisfies the inner Kan condition; i.e., has extension property with respect to all horn inclusions of trees. In particular, this condition guarantees we interpret sets X(T) as ‘sets operations’ parametrized by tree T, then there notion composition such operations (well-defined up homotopy) when grafting trees T and Tʹ. Section 12.1 i...

This survey provides an elementary introduction to operads and to their applications in homotopical algebra. The aim is to explain how the notion of an operad was prompted by the necessity to have an algebraic object which encodes higher homotopies. We try to show how universal this theory is by giving many applications in Algebra, Geometry, Topology, and Mathematical Physics. (This text is acc...

Diassociative algebras form a categoy of algebras recently introduced by Loday. A diassociative algebra is a vector space endowed with two associative binary operations satisfying some very natural relations. Any diassociative algebra is an algebra over the diassociative operad, and, among its most notable properties, this operad is the Koszul dual of the dendriform operad. We introduce here, b...

We consider four operads which are different varieties of the cacti operad. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant size one of the constituting loops. We show that both types of cacti are a semi–direct products as operads of their normalized versions with a re–scaling operad based on R>0 which are homotopic to direct produc...

The aim of this sequel to arXiv:1812.02935 is set up the cornerstones Koszul duality and Koszulity in context operads over a large class operadic categories. In particular, for these categories we will study concrete examples binary quadratic operads, describe their duals prove that they are Koszul. This includes whose algebras most important operad- PROP-like structures such as classical varia...

Minimal models of chain complexes associated with free torus actions on spaces have been extensively studied in the literature. In this paper, we discuss these constructions using language operads. The main goal paper is to define a new Koszul operad that has projections onto several operads used minimal model constructions.