Operads and Props

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 iterated loop spaces and homotopy invariant structures, but the theory of operads has recently received new inspiration from homological algebra, category theory, algebraic geometry and mathematical physics. The name operad and the formal definition appear first in the early 1970’s in J.P. May’s book [82], but a year or more earlier, M. Boardman and R. Vogt [8] described the same concept under the name categories of operators in standard form, inspired by PROPs and PACTs of Adams and Mac Lane [63]. As pointed out in [58], also Lambek’s definition of multicategory [56] (late 1960s) was almost equivalent to what is called today a coloured or many-sorted operad. Another important precursor was the associahedron K that appeared in J.D. Stasheff’s 1963 paper [97] on homotopy associativity of H-spaces. We do not, however, aspire to write an account on the history of operads here – we refer to the introduction of [79] or to [104] instead. Operads are important not in and of themselves but, like PROPs, through their representations, more commonly called algebras over operads or operad algebras. If an operad is thought of as a kind of algebraic theory, then an algebra over an operad is a model of that theory. Algebras over operads involve most of ‘classical’ algebras (associative, Lie, commutative associative, Poisson, &c.), loop spaces, moduli spaces of algebraic curves, vertex operator algebras, &c. Colored or many-sorted operads then describe diagrams of homomorphisms of these objects, homotopies between homomorphisms, modules, &c. PROPs generalize operads in the sense that they admit operations with several inputs and several outputs. Therefore various bialgebras (associative, Lie, infinitesimal) are PROPic algebras. PROPs were also used to encode ‘profiles’ of structures in formal differential geometry [84, 85]. By the renaissance of operads we mean the first half of the nineties of the last century when several papers which stimulated the rebirth of interest in operads appeared [27, 30, 37, 41, 43, 45, 68]. Let us mention the most important new ideas that emerged during this period. First of all, operads were recognized as the underlying combinatorial structure of the moduli space of stable algebraic curves in complex geometry, and of compactifications of configuration Date: January 6, 2006. The author was supported by the grant GA ČR 201/05/2117 and by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503. 1