Operads and PROPs
Section snippets
Operads
Although operads, operad algebras and most of related structures can be defined in an arbitrary symmetric monoidal category with countable coproducts, we decided to follow the choice of [58] and formulate precise definitions only for the category of modules over a commutative unital ring k, with the monoidal structure given by the tensor product over k. The reason for such a decision was to give, in Section 4, a clean construction of free operads. In a general monoidal
Non-unital operads
It turns out that the combinatorial structure of the moduli space of stable genus zero curves is captured by a certain non-unital version of operads. Let be the moduli space of -tuples of distinct numbered points on the complex projective line modulo projective automorphisms, that is, transformations of the form where with .
The moduli space has, for , a canonical compactification introduced by
Operad algebras
As we already remarked, operads are important through their representations called operad algebras or simply algebras.
Definition 23 Let V be a k-module and the endomorphism operad of V recalled in Example 2. A -algebra is a homomorphism of operads .
The above definition admits an obvious generalization for an arbitrary symmetric monoidal category with an internal hom-functor. The last assumption is necessary for the existence of the ‘internal’ endomorphism operad, see [83, Definition II.1.20].
Free operads and trees
The purpose of this section is three-fold. First, we want to study free operads because each operad is a quotient of a free one. The second reason why we are interested in free operads is that their construction involves trees. Indeed, it turns out that rooted trees provide ‘pasting schemes’ for operads and that, replacing trees by other types of graphs, one can introduce several important generalizations of operads, such as cyclic operads, modular operads, and PROPs. The last reason is that
Unbiased definitions
In this section, we review the definition of a triple (monad) and give, in Theorem 40, a description of unital and non-unital operads in terms of algebras over a triple. The relevant triples come from the endofunctors Ψ and Γ recalled in Section 4. Let be the strict symmetric monoidal category of endofunctors on a category where multiplication is the composition of functors.
Definition 38 A triple (also called a monad) T on a category is an associative and unital monoid in . The
Cyclic operads
In the following two sections we use the approach developed in Section 5 to introduce cyclic and modular operads. We recalled, in Example 14, the operad of Riemann spheres with parametrized labeled holes. Each was a right -space, with the operadic right -action permuting the labels of the holes . But each obviously admits a higher type of symmetry which interchanges the labels of all holes, including the label of the ‘output’ hole . Another
Modular operads
Let us consider again the -module of Riemann spheres with punctures. We saw that the operation of sewing the 0-th hole of the surface N to the i-th hole of the surface M defined on a cyclic operad structure. One may generalize this operation by defining, for , , , , the element by sewing the j-th hole of M to the i-th hole of N. Under this notation, . In the same manner, one may consider a single surface ,
PROPs
Operads are devices invented to describe structures consisting of operations with several inputs and one output. There are, however, important structures with operations having several inputs and several outputs. Let us recall the most prominent one.
Example 53 A (associative) bialgebra is a k-module V with a multiplication and a comultiplication (also called a diagonal) . The multiplication is associative: the comultiplication is coassociative: and the
Properads, dioperads and PROPs
As we saw in Proposition 33, under some mild assumptions, the components of free operads are finite-dimensional. In contrast, PROPs are huge objects. For example, the component of the free PROP used in the definition of the bialgebra PROP in Example 59 is infinite-dimensional for each , and also the components of the bialgebra PROP itself are infinite-dimensional, as follows from the fact that the Enriquez–Etingof basis (46) of has, for , infinitely many
Acknowledgements
I would like to express my gratitude to Ezra Getzler, Michiel Hazewinkel, Tom Leinster, Peter May, Jim Stasheff, Bruno Vallette and Rainer Vogt for careful reading the manuscript and useful suggestions. I am also indebted to the Institut des Hautes Études Scientifiques, Bures-sur-Yvette, for hospitality during the period when this article was completed.
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Supported by the grant GA ČR 201/05/2117 and by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.