Operads and Their Algebras — Mathematics Knowledge System

Overview

An operad P is a collection of objects P(n) (operations of arity n) with composition laws and symmetric group actions, encoding algebraic structures with multi-input operations. An algebra over P is an object A with 'evaluation maps' P(n) x A^n -> A satisfying the operad axioms. Classic examples: the associative operad Ass (P(n) = one point, encoding associative algebras), the commutative operad Comm (P(n) = one point with trivial symmetric action, encoding commutative algebras), and the Lie operad (encoding Lie algebras). Operads were invented to encode loop space structures (May, 1972) and are now central to deformation theory, homotopy theory, and mathematical physics.

Intuition

An operad captures 'operations with n inputs and 1 output' and how they compose. The associative operad Ass has one n-ary operation for each n (the product of n elements): the composition laws encode associativity. An algebra over Ass is an associative algebra. The commutative operad Comm additionally enforces commutativity. The little n-disks operad E_n encodes the structure on n-fold loop spaces: E_1 ~ Ass (1-fold loops are A-infinity), E_inf ~ Comm (infinite loop spaces are homotopy commutative). This hierarchy E_1, E_2, ..., E_inf measures how 'commutative' a multiplication is.

Formal Definition

Definition

A symmetric operad P in a symmetric monoidal category (C, otimes, I) consists of: objects P(n) for n >= 0; right actions of the symmetric group S_n on P(n); a unit morphism eta: I -> P(1); composition morphisms gamma: P(k) otimes P(n_1) otimes ... otimes P(n_k) -> P(n_1+...+n_k), subject to: associativity, unitality, and equivariance with respect to symmetric group actions. An algebra over P is an object A with maps P(n) otimes A^{otimes n} -> A satisfying compatibility with the operad structure.

P = \{P(n)\}_{n\ge 0},\quad P(n) \in C \text{ with } S_n\text{-action}

Operad data

\gamma: P(k) \otimes P(n_1) \otimes \cdots \otimes P(n_k) \to P(n_1+\cdots+n_k)

Composition

P(n) \otimes A^{\otimes n} \to A \quad (\text{algebra action})

Algebra over P

E_1 \simeq \mathrm{Ass},\quad E_\infty \simeq \mathrm{Comm} \quad (\text{up to homotopy})

Little disks operad hierarchy

Notation

Notation	Meaning
$P(n)$	Space/object of n-ary operations in operad P
$E_n$	Little n-disks (or n-cubes) operad
$P \,!\, Q$	Koszul dual operad of P
$P_\infty$	Homotopy version (cofibrant resolution) of P

Theorems

Theorem 1: May Recognition Theorem

A connected topological space X is (weakly homotopy equivalent to) an n-fold loop space Omega^n Y if and only if it is an algebra over the little n-disks operad E_n. For n=infinity: X is an infinite loop space iff it is an E-infinity algebra (Omega^inf Sigma^inf X -- Segal's delooping machine).

Theorem 2: Koszul Duality of Operads

For a quadratic operad P (generated in arity 2 with quadratic relations), there is a Koszul dual operad P^! and a natural chain complex of operads (the bar complex). P is Koszul if its bar complex is acyclic (a resolution). Koszul pairs: Ass^! = Ass (self-dual); Comm^! = Lie; Lie^! = Comm. A Lie-infinity algebra (L-infinity or sh Lie algebra) is an algebra over Lie_infinity = the cofibrant replacement of Lie. Koszul duality governs deformation theory: deformations of a Comm-algebra structure are controlled by a Lie-infinity algebra.

Theorem 3: Deligne's Conjecture (now Theorem)

The Hochschild cochain complex C*(A, A) of an associative algebra A carries a natural action of the E_2-operad (little 2-disks). This gives the Hochschild complex the structure of a homotopy Gerstenhaber algebra. Proved by Kontsevich-Soibelman, McClure-Smith, Berger-Fresse, and others in 2000-2003. The conjecture originated from Deligne's 1993 letter.

Worked Examples

1
Define Ass: Ass(n) = k[S_n] (the group algebra of the symmetric group on n elements, as a right S_n-module). Alternatively (over a field): Ass(n) = {one point} with the regular S_n-action, i.e., Ass(n) as a set = S_n, and S_n acts by right multiplication.
2
Composition in Ass: given sigma in Ass(k) and tau_1,...,tau_k in Ass(n_1),...,Ass(n_k), gamma(sigma; tau_1,...,tau_k) = sigma(tau_1,...,tau_k) (block permutation). This is the composition of permutations under block substitution.
3
An Ass-algebra: a vector space A with maps mu_n: Ass(n) otimes A^{otimes n} -> A. The key map is mu_2: A otimes A -> A (multiplication) from Ass(2) = {id, (12)} and mu_n are determined by mu_2 via associativity.
4
The operad axioms force mu_2 to be associative: gamma in Ass forces (a*b)*c = a*(b*c). So Ass-algebras = associative algebras.
$\text{Ass-algebras} = \text{associative algebras}$

✓ Answer

Ass(n) = k[S_n] with block composition. An Ass-algebra is an associative algebra with multiplication mu_2 and higher operations determined by associativity.

Practice Problems

Hardfree response

State the Koszul dual pair (Comm, Lie) and explain what an L-infinity algebra is.

Common Mistakes

Common Mistake

Confusing an operad with its algebras.

An operad P is the 'theory' (the package of operations and their composition rules). An algebra over P is a specific structure (an object A with operations P(n) x A^n -> A satisfying the axioms). The operad Ass is NOT an associative algebra; rather, an algebra over Ass is an associative algebra. Similarly, the Lie operad is not itself a Lie algebra; a Lie algebra is an algebra over the Lie operad. This distinction is like the difference between a group (theory) and a group representation (model).

Quiz

The little n-disks operad E_n encodes:

The structure on n-fold products of abelian groupsThe structure naturally present on n-fold loop spacesThe structure on spaces with n commuting differentialsThe Koszul dual of the commutative operad

Historical Background

Peter May coined the term 'operad' in his 1972 book 'The Geometry of Iterated Loop Spaces', designed to encode the structure maps of iterated loop spaces. The concept was anticipated by Boardman and Vogt (1973) and related work. Jim Stasheff had studied A-infinity structures (homotopy associativity) via associahedra in 1963. The algebraic theory of operads was systematized by Ginzburg and Kapranov (1994) who introduced the Koszul duality of operads. Loday and Vallette's book (2012) is now the standard reference. Operads appear in: string field theory, Kontsevich's formality theorem, Deligne's conjecture (proved), and the theory of En-algebras.

1963
Stasheff introduces A-infinity structures (homotopy associativity) via associahedra
James Stasheff
1972
May defines operads to encode iterated loop space structures
J. Peter May
1994
Ginzburg-Kapranov introduce Koszul duality for operads
Victor Ginzburg, Mikhail Kapranov
1997
Kontsevich uses operads for deformation quantization and formality theorem
Maxim Kontsevich

Summary

Operad P: collection P(n) of n-ary operations with S_n-actions and composition laws.
Algebra over P: object A with maps P(n) otimes A^n -> A compatible with operad structure.
Classic examples: Ass (associative algebras), Comm (commutative algebras), Lie (Lie algebras).
E_n operads: structure on n-fold loop spaces; Koszul duality: Comm^! = Lie.

References

BookMay, J.P. The Geometry of Iterated Loop Spaces. Springer, 1972.
BookLoday, J.L. and Vallette, B. Algebraic Operads. Springer, 2012.
WebsitenLab -- Operad