We review definitions and basic properties of operads, PROPs and algebras over these structures. Dedicated to the memory of Jakub Jan Ryba (1765–1815) Operads involve an abstraction of the family {Map(X, X)}n≥0 of composable functions of several variables together with an action of permutations of variables. As such, they were originally studied as a tool in homotopy theory, specifically for it...

A new hierarchy of operads over the linear spans $$\delta $$ -cliffs, which are some words integers, is introduced. These intended to be analogues operad permutations, also known as associative symmetric operad. We obtain whose partial compositions can described in terms intervals lattice -cliffs. very peculiar world combinatorial since, despite relative simplicity for their construction, they ...

We construct explicit minimal models for the (hyper)operads governing modular, cyclic and ordinary operads, wheeled properads, respectively. Algebras these are homotopy versions of corresponding structures.

We study homotopy-coherent commutative multiplicative structures on equivariant spaces and spectra. We define N∞ operads, equivariant generalizations of E∞ operads. Algebras in equivariant spectra over an N∞ operad model homotopically commutative equivariant ring spectra that only admit certain collections of Hill-Hopkins-Ravenel norms, determined by the operad. Analogously, algebras in equivar...

We decribe the correspondence between normalised ω-operads in the sense of [1] and certain lax monoidal structures on the category of globular sets. As with ordinary monoidal categories, one has a notion of category enriched in a lax monoidal category. Within the aforementioned correspondence, we provide also an equivalence between the algebras of a given normalised ωoperad, and categories enri...

In §1.4, we briefly explained the definition of a natural model structure for simplicial (and topological) operads. In what follows, we also refer to this model structure as the projective model structure. The weak-equivalences (respectively, fibrations) are, according to this definition, the morphisms of operads which form a weak-equivalence (respectively, a fibration) in the base category of ...

Informally, a homotopy monoid is a monoid-like structure in which properties such as associativity only hold ‘up to homotopy’ in some consistent way. This short paper comprises a rigorous definition of homotopy monoid and a brief analysis of some examples. It is a much-abbreviated version of the paper ‘Homotopy Algebras for Operads’ (math.QA/0002180), and does not assume any knowledge of operads.

Let M be a bicomplete, closed symmetric monoidal category. Let P be an operad in M, i.e., a monoid in the category of symmetric sequences of objects in M, with its composition monoidal structure. Let R be a P-co-ring, i.e., a comonoid in the category of P-bimodules. The co-ring R induces a natural “fattening” of the category of P-(co)algebras, expanding the morphism sets while leaving the objec...

We extend the W-construction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for well-pointed Σ-cofibrant operads. The standard simplicial resolution of Godement as well as the cobar-bar chain resolution are shown to be particular instances of this generalised W-construction.