Algebra+Homotopy=Operad

Abstract

This survey provides an elementary introduction to operads and to their applications in homotopical algebra. The aim is to explain how the notion of an operad was prompted by the necessity to have an algebraic object which encodes higher homotopies. We try to show how universal this theory is by giving many applications in Algebra, Geometry, Topology, and Mathematical Physics. (This text is accessible to any student knowing what tensor products, chain complexes, and categories are.)

59 figures, 31 exercices, 43 pages

References in corpus (11)

• Deformation quantization of Poisson manifolds, I
• Operads, homotopy algebra and iterated integrals for double loop spaces
• Algebraic Structures and Differential Geometry in 2D String Theory
• Combinatorial operad actions on cochains
• Another proof of M. Kontsevich formality theorem
• Courant Algebroids and Strongly Homotopy Lie Algebras
• Nonhomogeneous quadratic duality and curvature
• Homological perturbation theory for algebras over operads
• Coloured Koszul duality and strongly homotopy operads
• Dyer-Lashof-Cohen operations in Hochschild cohomology
• Free frobenius algebra on the differential forms of a manifold

Cited by in corpus (19)

• The Frobenius properad is Koszul
• The S-Matrix of superstring field theory
• Infinity Operads and Monoidal Categories with Group Equivariance
• Perspectives on the pure spinor superfield formalism
• The rational homotopy type of (n-1)-connected manifolds of dimension up to 5n-3
• Gauge Invariant Perturbation Theory via Homotopy Transfer
• The Golod property for Stanley-Reisner rings in varying characteristic
• A period map for global derived stacks
• Twisted IBL-infinity-algebra and string topology: First look and examples
• Tree- versus graph-level quasilocal Poincare duality on S^1
• IBL-Infinity Model of String Topology from Perturbative Chern-Simons Theory
• Topological Perspectives on Statistical Quantities I
• Koszul duality for Iwasawa algebras modulo p
• Concerning the ghost contribution to the one-loop integrands in open string field theory
• $E_2{L}_{\infty }$-algebras, Generalized Geometry, and Tensor Hierarchies
• Formality of spaces with Lusternik-Schnirelmann category 1
• Non-formality in $\mathrm{Pin}(2)$-monopole Floer homology
• Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology
• Exact triangles as modules over an $A_{\infty }$-category