Several topological and homological operads based on families of projectively weighted arcs in bounded surfaces are introduced and studied. The spaces underlying the basic operad are identified with open subsets of a combinatorial compactification due to Penner of a space closely related to Riemann's moduli space. Algebras over these operads are shown to be Batalin-Vilkovisky algebras, where the entire BV structure is realized simplicially. Furthermore, our basic operad contains the cacti operad up to homotopy. New operad structures on the circle are classified and combined with the basic operad to produce geometrically natural extensions of the algebraic structure of BV algebras, which are also computed.

In this paper we prove a "Leray theorem" for pre-Lie algebras. We define a notion of "Hopf" pre-Lie algebra: it is a pre-Lie algebra together with a non-associative permutative coproduct ∆ and a compatibility relation between the pre-Lie product and the coproduct ∆. A non-associative permutative algebra is a vector space together with a product satisfying the relation (ab)c = (ac)b. A non-associative permutative coalgebra is the dual notion. We prove that any connected "Hopf" pre-Lie algebra is a free pre-Lie algebra. It uses the description of pre-Lie algebras in terms of rooted trees developed by Chapoton and the author. We also interpret this theorem by way of cogroups in the category of pre-Lie algebras.

Starting from an operad, one can build a family of posets. From this family of posets, one can define an incidence Hopf algebra. By another construction, one can also build a group directly from the operad. We then consider its Hopf algebra of functions. We prove that there exists a surjective morphism from the latter Hopf algebra to the former one. This is illustrated by the case of an operad built on rooted trees, the NAP operad, where the incidence Hopf algebra is identified with the Connes-Kreimer Hopf algebra of rooted trees.where I is a finite set and ≃ runs over the set of equivalence relations on I. Note that this monoidal functor is not symmetric.The data of a species P is equivalent to the data of a collection of sets P(n) with actions of the symmetric groups. The set P(n) can be defined as P ({1, . . . , n}), with the obvious action of the symmetric group S n . The other way round, one can recover the set P(I) as a colimit. Set-operadsA set-operad P is a monoid with unit in the monoidal category of species for the tensor product •. This means the data of a morphism of species γ : P • P → P,

Abstract. In this paper we study the homology of 2 versions of the swiss-cheese operad. We prove that the zeroth homology of these two versions are Koszul operads and relate this to strong homotopy Lebiniz pairs and OCHA, defined by Kajiura and Stasheff in [13].