12.2 A-infinity algebras and operads

A-infinity algebras expand on associative algebras, allowing for higher homotopies. They consist of graded vector spaces with multilinear maps satisfying generalized associativity conditions. The Stasheff associahedron encodes these relations, with faces corresponding to the maps in an A-infinity algebra.

Operads provide a framework for studying algebraic structures with multiple operations. They consist of sets representing operations with composition maps. The bar construction assigns differential graded coalgebras to operads, while Koszul duality establishes correspondences between certain operad pairs.

A-infinity Algebras and Homotopy

Generalizing Associative Algebras

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• A-infinity algebras generalize the notion of associative algebras by allowing for higher homotopies
• Consist of a graded vector space $A$ equipped with a family of multilinear maps $m_n: A^{\otimes n} \to A$ of degree $2-n$ for each $n \geq 1$
• The maps $m_n$ satisfy a sequence of relations that generalize the associativity condition for ordinary algebras
• The first relation states that $m_1$ is a differential, meaning $m_1^2 = 0$, and the second relation is the Leibniz rule for $m_2$ with respect to $m_1$

Higher Homotopies and the Stasheff Associahedron

• Higher homotopies in an A-infinity algebra are encoded by the maps $m_n$ for $n \geq 3$
• These higher homotopies measure the failure of associativity up to homotopy
• The Stasheff associahedron is a polytope whose vertices correspond to the different ways of bracketing $n$ elements
• The faces of the associahedron correspond to the relations satisfied by the maps $m_n$ in an A-infinity algebra
  • For example, the pentagonal face corresponds to the relation involving $m_2$ and $m_3$

Transferring A-infinity Structures

• The homotopy transfer theorem states that an A-infinity structure can be transferred along a homotopy equivalence
• More precisely, if $A$ and $B$ are homotopy equivalent chain complexes and $A$ has an A-infinity structure, then $B$ inherits an A-infinity structure as well
• This theorem is useful for constructing A-infinity algebras from simpler data
• For instance, it can be used to transfer an A-infinity structure from a differential graded algebra to its homology

Operads and Bar Construction

Operads as a Unified Framework

• Operads provide a unified framework for studying algebraic structures with multiple operations
• An operad $\mathcal{O}$ consists of a collection of sets $\mathcal{O}(n)$ for each $n \geq 0$, representing operations with $n$ inputs
• These sets are equipped with composition maps that allow for substitution of operations into each other
• Operads encode many common algebraic structures, such as associative algebras, commutative algebras, and Lie algebras
  • For example, the associative operad $\mathcal{A}ss$ has $\mathcal{A}ss(n) = \Sigma_n$, the symmetric group on $n$ letters

The Bar Construction

• The bar construction is a functor that assigns to each operad $\mathcal{O}$ a differential graded coalgebra $B(\mathcal{O})$
• It is defined as the free coalgebra generated by the suspension of $\mathcal{O}$, with a differential induced by the composition maps
• The bar construction is a key tool in the study of operads and their algebras
• For instance, the bar construction of an A-infinity algebra is a differential graded coalgebra, and morphisms of A-infinity algebras correspond to coalgebra morphisms between their bar constructions

Koszul Duality for Operads

• Koszul duality is a correspondence between certain pairs of operads
• An operad $\mathcal{O}$ is Koszul if its bar construction is quasi-isomorphic to the suspension of another operad $\mathcal{O}^!$, called the Koszul dual of $\mathcal{O}$
• Koszul duality provides a way to study an operad $\mathcal{O}$ in terms of its Koszul dual $\mathcal{O}^!$, which often has a simpler structure
• Many important operads, such as the associative, commutative, and Lie operads, are Koszul
  • For example, the Koszul dual of the associative operad is the shifted Lie operad

Minimal Models

Minimal A-infinity Algebras

• A minimal A-infinity algebra is an A-infinity algebra $(A, \{m_n\})$ where $m_1 = 0$
• In other words, the differential on the underlying chain complex of $A$ vanishes
• Minimal A-infinity algebras are important because they serve as "simplified models" for general A-infinity algebras
• Every A-infinity algebra is quasi-isomorphic to a minimal one, called its minimal model

Constructing Minimal Models

• The existence of minimal models for A-infinity algebras is a consequence of the homotopy transfer theorem
• Given an A-infinity algebra $(A, \{m_n\})$, one can construct a minimal model by the following steps:
  1. Choose a homotopy equivalence between $A$ and its homology $H(A)$
  2. Transfer the A-infinity structure from $A$ to $H(A)$ using the homotopy transfer theorem
• The resulting A-infinity algebra on $H(A)$ is a minimal model for $A$
• The structure maps of the minimal model can be described explicitly in terms of the homotopy equivalence data

Applications of Minimal Models

• Minimal models are a powerful tool for studying the homotopy theory of A-infinity algebras
• They provide a way to reduce questions about general A-infinity algebras to questions about simpler, minimal ones
• For example, two A-infinity algebras are quasi-isomorphic if and only if their minimal models are isomorphic
• Minimal models also play a key role in the construction of spectral sequences that compute the Hochschild cohomology of A-infinity algebras
  • These spectral sequences are analogous to the classical Adams spectral sequence in algebraic topology