Operads

J. Peter May in 1972 proved that n-fold loop spaces are precisely algebras over the little n-disks operad. Fifty years later, E_n-algebras became the central language of derived algebraic geometry and deformation theory.

• Topological quantum field theory: D-brane categories are E_n-algebras in derived categories
• Deformation theory: L-infinity and A-infinity algebras are algebras over the corresponding operads
• Algebraic geometry: Lurie's factorization algebras in derived algebraic geometry rest on E_n-operads

Предварительные знания

• Previous lesson

Definition of an Operad

Operads were introduced by J. Peter May in 1972 to encode the algebraic structure of iterated loop spaces, and rediscovered by Loday and others in the 1990s. Today operads classify the homotopy types of E_n-algebras (Lurie, 2009) and underlie the formality theorems used in deformation quantization (Kontsevich, 1997).

An operad P is a collection {P(n)}_{n>=0} of right S_n-modules, with unit eta: k -> P(1) and composition maps gamma: P(k) ⊗ P(n_1) ⊗ ... ⊗ P(n_k) -> P(n_1+...+n_k) satisfying associativity, unitality, and equivariance. A P-algebra is an object A with action maps P(n) ⊗ A^{⊗n} -> A compatible with gamma.

Little Disks Operad and Loop Spaces

The topological little n-disks operad D_n: D_n(k) = configuration space of k non-overlapping disks in D^n. May's recognition theorem (1972): X is an n-fold loop space if and only if X is a group-like D_n-algebra. This gives a complete algebraic characterization of loop spaces.

Key Ideas

• Operad P: {P(n)} with S_n-action, unit, and associative composition gamma
• dim Ass(n) = n!, dim Com(n) = 1, dim Lie(n) = (n-1)! -- the classical cases
• Koszul duality: Lie^! = Com, Com^! = Lie, Ass^! = Ass
• D_n = little n-disks operad: D_n(k) = configurations of k disks
• May's theorem: X is an Omega^n-space iff X is a group-like D_n-algebra

Further Directions

These ideas open paths to deeper mathematics.

• ct-28-model-categories — extends

Вопросы для размышления

• Give a concrete example.
• How does this connect to other areas of mathematics?

Связанные уроки

• aa-12-modules
• plt-11-typeclasses