Operads and PROPs

doi:10.1016/S1570-7954(07)05002-4

Handbook of Algebra

Volume 5, 2008, Pages 87-140

Dedicated to the memory of Jakub Jan Ryba (1765–1815)

https://doi.org/10.1016/S1570-7954(07)05002-4 Get rights and content

Abstract

We review definitions and basic properties of operads, PROPs and algebras over these structures.

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 ${Mod}_{k} = ({Mod}_{k}, \otimes)$ of modules over a commutative unital ring k, with the monoidal structure given by the tensor product $\otimes = \otimes_{k}$ 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 $M_{0, n + 1}$ be the moduli space of $(n + 1)$ -tuples $(x_{0}, \dots, x_{n})$ of distinct numbered points on the complex projective line ${CP}^{1}$ modulo projective automorphisms, that is, transformations of the form ${CP}^{1} ∋ [ξ_{1}, ξ_{2}] \mapsto [a ξ_{1} + b ξ_{2}, c ξ_{1} + d ξ_{2}] \in {CP}^{1},$ where $a, b, c, d \in C$ with $a d - b c \neq 0$ .

The moduli space $M_{0, n + 1}$ has, for $n ⩾ 2$ , a canonical compactification ${\bar{M}}_{0} (n) \supset M_{0, n + 1}$ 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 ${E nd}_{V}$ the endomorphism operad of V recalled in Example 2. A $P$ -algebra is a homomorphism of operads $ρ : P \to {E nd}_{V}$ .

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 $End (C)$ be the strict symmetric monoidal category of endofunctors on a category $C$ where multiplication is the composition of functors.

Definition 38

A triple (also called a monad) T on a category $C$ is an associative and unital monoid $(T, μ, υ)$ in $End (C)$ . 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 ${\hat{M}}_{0} = {{\hat{M}}_{0} (n)}_{n ⩾ 0}$ of Riemann spheres with parametrized labeled holes. Each ${\hat{M}}_{0} (n)$ was a right $Σ_{n}$ -space, with the operadic right $Σ_{n}$ -action permuting the labels $1, \dots, n$ of the holes $u_{1}, \dots, u_{n}$ . But each ${\hat{M}}_{0} (n)$ obviously admits a higher type of symmetry which interchanges the labels $0, \dots, n$ of all holes, including the label of the ‘output’ hole $u_{0}$ . Another

Modular operads

Let us consider again the $Σ^{+}$ -module ${\hat{M}}_{0} = {{\hat{M}}_{0} (n)}_{n ⩾ 0}$ of Riemann spheres with punctures. We saw that the operation $M, N \mapsto M ○_{i} N$ of sewing the 0-th hole of the surface N to the i-th hole of the surface M defined on ${\hat{M}}_{0}$ a cyclic operad structure. One may generalize this operation by defining, for $M \in {\hat{M}}_{0} (m)$ , $N \in {\hat{M}}_{0} (n)$ , $0 ⩽ i ⩽ m$ , $0 ⩽ j ⩽ n$ , the element $M {}_{i}○_{j} N \in {\hat{M}}_{0} (m + n - 1)$ by sewing the j-th hole of M to the i-th hole of N. Under this notation, $○_{i} = {}_{i}○_{0}$ . In the same manner, one may consider a single surface $M \in {\hat{M}}_{0} (n)$ ,

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 $μ : V \otimes V \to V$ and a comultiplication (also called a diagonal) $Δ : V \to V \otimes V$ . The multiplication is associative: $μ (μ \otimes {id}_{V}) = μ ({id}_{V} \otimes μ),$ the comultiplication is coassociative: $(Δ \otimes {id}_{V}) Δ = ({id}_{V} \otimes Δ) Δ$ and the

Properads, dioperads and $\frac{1}{2}$ 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 $Γ_{P} (,) (m, n)$ of the free PROP $Γ_{P} (,)$ used in the definition of the bialgebra PROP $B$ in Example 59 is infinite-dimensional for each $m, n ⩾ 1$ , and also the components of the bialgebra PROP $B$ itself are infinite-dimensional, as follows from the fact that the Enriquez–Etingof basis (46) of $B (m, n)$ has, for $m, n ⩾ 1$ , 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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¹: Supported by the grant GA ČR 201/05/2117 and by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.

View full text

Operads and PROPs

Abstract

Section snippets

Operads

Non-unital operads

Operad algebras

Free operads and trees

Unbiased definitions

Cyclic operads

Modular operads

PROPs

Properads, dioperads and 12PROPs

Acknowledgements

J. Pure Appl. Algebra

Adv. Math.

J. Pure Appl. Algebra

Topology

J. Pure Appl. Algebra

J. Algebra

J. Pure Appl. Algebra

J. Algebra

The Eckmann–Hilton argument, higher operads and En-spaces

The combinatorics of iterated loop spaces

Chiral Algebras

Axiomatic homotopy theory for operads

Comment. Math. Helv.

Resolution of coloured operads and rectification of homotopy algebras

Homotopy-everything H-spaces

Bull. Amer. Math. Soc.

Homotopy Invariant Algebraic Structures on Topological Spaces

T-catégories (catégories dans un triple)

Cah. Topol. Géom. Différ.

Homologie cycliques: rapport sur des travaux récents de Connes, Karoubi, Loday, Quillen ...

Sémin. Bourbaki Exp. no. 621, 1983–1984

Astérisque

String topology

Notes on string topology

Renormalization in quantum field theory and the Riemann–Hilbert problem I: The Hopf algebra structure of graphs and the main theorem

Comm. Math. Phys.

Resumé des premiérs exposés de A. Grothendieck

The irreducibility of the space of curves of given genus

Inst. Hautes Études Sci. Publ. Math.

Tessellations of moduli spaces and the mosaic operad

Adjoint functors and triples

Illinois J. Math.

The development of structured ring spectra

Modern foundations of stable homotopy theory

Rings, Modules, and Algebras in Stable Homotopy Theory

On the invertibility of quantization functors

The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points

Distributive laws, bialgebras, and cohomology

Koszul duality of operads and homology of partition posets

Koszul duality for dioperads

Math. Res. Lett.

The cohomology structure of an associative ring

Ann. of Math.

Operads and moduli spaces of genus 0 Riemann surfaces

Operads, homotopy algebra, and iterated integrals for double loop spaces

Cyclic operads and cyclic homology

Modular operads

Compos. Math.

Koszul duality for operads

Duke Math. J.

On perturbations and A∞-structures

Bull. Soc. Math. Belg.

Lectures on minimal models

Mem. Soc. Math. France (N.S.) 9–10

Algebraic Geometry

A Course in Homological Algebra

Homological algebra of homotopy algebras

Comm. Algebra

Tamarkin's proof of Kontsevich formality theorem

Forum Math.

Homotopy Lie algebras

Adv. Soviet Math.

Cyclic operads and algebra of chord diagrams

Selecta Math. (N.S.)

Higher string topology on general spaces

On Kontsevich's Hochschild cohomology conjecture

Operadic formulation of topological vertex algebras and Gerstenhaber or Batalin–Vilkovisky algebras

Properads, dioperads and $\frac{1}{2}$ PROPs

The Eckmann–Hilton argument, higher operads and $E_{n}$ -spaces

On perturbations and $A_{\infty}$ -structures