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Generalization of associativity properties
specification of how to compose these operations. Given an operad O {\displaystyle O} , one defines an algebra over O {\displaystyle O} to be a set together with
Operad
algebra, an operad algebra is an "algebra" over an operad. It is a generalization of an associative algebra over a commutative ring R, with an operad
Operad_algebra
In mathematics, an A∞-operad is a type of operad used in algebraic topology and homotopy theory to describe algebraic structures where the property of
A∞-operad
In mathematics, the Lie operad is an operad whose algebras are Lie algebras. The notion (at least one version) was introduced by Ginzburg & Kapranov (1994)
Lie_operad
Infinite loop space
In the theory of operads in algebra and algebraic topology, an E∞-operad is a parameter space for a multiplication map that is associative and commutative
E∞-operad
planar operad is the set of all the planar tangles (up to isomorphism) with such compositions. A planar algebra is a representation of the planar operad; more
Planar_algebra
Theory of algebraic structures in general
development in category theory is operad theory – an operad is a set of operations, similar to a universal algebra, but restricted in that equations are
Universal_algebra
Various mathematical dualites
algebra over an operad is an object on which these n-ary operations act. For example, there is an operad called the associative operad whose algebras
Koszul_duality
Vector space equipped with a bilinear product
like an algebra over a field. Algebra over an operad Alternative algebra Clifford algebra Composition algebra Differential algebra Free algebra Geometric
Algebra_over_a_field
American mathematician (born 1939)
J. Peter May at the Mathematics Genealogy Project May, J. Peter. "Operads, Algebras, and Modules" (PDF). math.uchicago.edu. p. 2. Retrieved 28 September
J._Peter_May
Algebra used in 2D conformal field theories and string theory
formulations of the vertex algebra axioms include Borcherds's later work on singular commutative rings, algebras over certain operads on curves introduced by
Vertex_operator_algebra
definition of a homotopy algebra using the theory of operads: that is, a homotopy Lie algebra is an algebra over an operad in the category of chain complexes
Homotopy_Lie_algebra
Algebraic structure used in analysis
of a Lie algebra Leibniz algebra Lie algebra cohomology Lie algebra extension Lie algebra representation Lie bialgebra Lie coalgebra Lie operad Particle
Lie_algebra
the free Lie algebra on the components of the link, as discussed in that article. See also Lie operad for the use of a free Lie algebra in the construction
Free_Lie_algebra
Operation in graph theory
9: 30–43, MR 0209178. Weber, Mark (2013), "Free products of higher operad algebras", Theory and Applications of Categories, 28 (2): 24–65. Weisstein,
Cartesian_product_of_graphs
Concept in universal algebra in mathematics
between Lawvere theories and abstract clones. Term algebra Operad Denecke, Klaus (2003). "Menger algebras and clones of terms". East–West Journal of Mathematics
Clone_(algebra)
Branch of mathematics
derived algebraic geometry. Derived scheme Pursuing Stacks Noncommutative algebraic geometry Simplicial commutative ring Derivator Algebra over an operad En-ring
Derived_algebraic_geometry
(2012-02-15). "Algebra+Homotopy=Operad". arXiv:1202.3245 [math.AT]. Allocca, Michael; Lada, Thomas. "A Finite Dimensional A-infinity algebra example" (PDF)
Homotopy_associative_algebra
Sum of terms, each multiplied with a scalar
More abstractly, in the language of operad theory, one can consider vector spaces to be algebras over the operad R ∞ {\displaystyle \mathbf {R} ^{\infty
Linear_combination
arXiv:math/0001151. Getzler, Ezra; Jones, J. D. S. (1994). "Operads, homotopy algebra and iterated integrals for double loop spaces". arXiv:hep-th/9403055
Deligne's conjecture on Hochschild cohomology
Deligne's_conjecture_on_Hochschild_cohomology
Branch of mathematics
conjecture about affine space.) In this line of the approach, the notion of operad, a set or space of operations, becomes prominent: in the introduction to
Noncommutative algebraic geometry
Noncommutative_algebraic_geometry
Algebraic structure in homological algebra
homological algebra, algebraic topology, and algebraic geometry – a differential graded algebra (or DGA, or DG algebra) is an algebraic structure often
Differential_graded_algebra
In mathematics, a (right) Leibniz algebra, named after Gottfried Wilhelm Leibniz, sometimes called a Loday algebra, after Jean-Louis Loday, is a module
Leibniz_algebra
Cohomology class
that ΩX is an algebra over the little interval operad. This is an example of an A ∞ {\displaystyle A_{\infty }} -operad, i.e. an operad of topological
Highly structured ring spectrum
Highly_structured_ring_spectrum
"Nilpotency of Zinbiel algebras". J. Dyn. Control Syst. 11 (2): 195–213. Ginzburg, Victor; Kapranov, Mikhail (1994). "Koszul duality for operads". Duke Mathematical
Zinbiel_algebra
Symmetric monoidal infinity category
definition is that A is an algebra in C over the little n-disks operad. An E n {\displaystyle {\mathcal {E}}_{n}} -algebra in vector spaces over a field
En-ring
Generalization of the concept of category that allows morphisms of multiple arity
several variables. Multicategories are also sometimes called operads, or colored operads. A (non-symmetric) multicategory consists of a collection (often
Multicategory
2-category version of algebra
of algebra for T, that satisfies the laws up to coherent isomorphisms. Operad Shulman, Michael A. (2012). "Not every pseudoalgebra is equivalent to a
Pseudoalgebra
{\displaystyle k[x]} . Loday, Jean-Louis; Vallette, Bruno (2012). Algebraic operads. Grundlehren der Mathematischen Wissenschaften. Vol. 346. Berlin:
Augmentation_(algebra)
generators. Chapoton, F.; Livernet, M. (2001), "Pre-Lie algebras and the rooted trees operad", International Mathematics Research Notices, 2001 (8): 395–408
Pre-Lie_algebra
Branch of topology
cacti operad". String topology and cyclic homology. Basel: Birkhäuser. ISBN 978-3-7643-7388-7. Getzler, Ezra (1994). "Batalin-Vilkovisky algebras and two-dimensional
String_topology
Mathematical group
ISBN 978-0-521-59642-8, MR 1483118 Fresse, Benoit (2017), Homotopy of Operads and Grothendieck-Teichmüller Groups: Part 2: The Applications of (Rational)
Grothendieck–Teichmüller group
Grothendieck–Teichmüller_group
Generalization of a semialgebraic set
S2CID 50187913. Maxim Kontsevich and Yan Soibelman. “Deformations of algebras over operads and the Deligne conjecture”. In: Conférence Moshé Flato 1999, Vol
Piecewise_algebraic_space
Type of monoidal category in category theory
(2002). Operads in Algebra, Topology and Physics. American Mathematical Society. ISBN 978-0-8218-4362-8. Leinster, Tom (2004). Higher Operads, Higher
PROP_(category_theory)
Operation in algebra and mathematics
monads, monoidal approach to bicategorical structures and generalized operads". Theory and Applications of Categories. 30: 332–386. doi:10.70930/tac/nfe2xf1p
Monad_(category_theory)
Polytope of operations
presheaf on the category of opetopes. higher-order operad Leinster, Tom (2004). "Opetopes". Higher Operads, Higher Categories. London Mathematical Society
Opetope
Kazakh mathematician and physicist (born 1956)
Journal of Algebraic Combinatorics. – 2011. – V. 33, No.4. – P. 531–542. Dzhumadildaev A.S., Codimension growth and non-Koszulity of Novikov operad // Communications
Askar_Dzhumadildayev
Moerdijk & Weiss (2007). They have the same relation to (colored symmetric) operads, also called symmetric multicategories, that simplicial sets have to categories
Dendroidal_set
Beilinson and V. Drinfeld in their book "Chiral algebras". The notion can also be defined as a colored operad or multicategory. In particular, a pseudo-tensor
Pseudo-tensor_category
Process in mathematics
337–350, MR 1767430. Loday, Jean-Louis; Vallette, Bruno (2012), Algebraic operads (PDF), Grundlehren der Mathematischen Wissenschaften [Fundamental
Koszul_algebra
Shnider and Stasheff Operads in algebra, topology, and physics was the first book to provide a systematic treatment of operad theory, an area of mathematics
Steve_Shnider
Mathematics glossary
glossary of properties and concepts in algebraic topology in mathematics. See also: glossary of topology, list of algebraic topology topics, glossary of category
Glossary of algebraic topology
Glossary_of_algebraic_topology
Mathematical operation with two operands
Binary operations are the keystone of most structures that are studied in algebra, in particular in semigroups, monoids, groups, rings, fields, and vector
Binary_operation
to finding a (quantum) algebra whose classical limit is a given (classical) algebra such as a Lie algebra or a Poisson algebra. Intuitively, a deformation
Deformation_quantization
Gröbner bases for non-commutative algebra
for: For power series algebras. For certain quiver Hecke algebras. For category algebras. For small categories. For shuffle operads. The lemma has been
Bergman's_diamond_lemma
Mathematical concept
Henri Cartan, Samuel Eilenberg, Homological algebra Fresse, Benoit (21 April 2017). Homotopy of Operads and Grothendieck-Teichmuller Groups. Mathematical
Differential_graded_module
linear algebra Contents: Top A B C D E F G H I J K L M N O P Q R S T U V W X Y Z See also References Operad theory a type of abstract algebra concerned
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Russian-American mathematician
Hochschild cohomology, cohomology of vertex operator algebras, and string topology (see cactus operad). He is a Fellow of the American Mathematical Society
Alexander_A._Voronov
algebraic notions have previously been generalized to dialgebras. Dialgebra also attempts to obtain Lie algebras from associated algebras. F-algebra Lambek
Dialgebra
History of maths
Categories of abstract algebraic structures including representation theory and universal algebra; Homological algebra; Homotopical algebra; Topology using categories
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
(under construction) Lecture 4 of Part II of Moerdijk-Toen, Simplicial Methods for Operads and Algebraic Geometry Ch. 2.2. of Toen-Vezzosi's HAG II v t e
Derived_tensor_product
Bosnian-American mathematician
Mathematical Social Sciences, 114 (2021), 39–57 "Formality of the little N-discs operad", with P. Lambrechts, Memoirs of the American Mathematical Society, 239
Ismar_Volić
Russian American mathematician (born 1957)
theory for operads. In noncommutative geometry, Ginzburg defined, following earlier ideas of Maxim Kontsevich, the notion of Calabi–Yau algebra. An important
Victor_Ginzburg
"colored operad" is more primitive than that of operad: in fact, an operad can be defined as a colored operad with a single object. comma Given functors f
Glossary_of_category_theory
General theory of mathematical structures
Saunders Mac Lane in the mid-20th century in their foundational work on algebraic topology. Category theory can be used in most areas of mathematics. In
Category_theory
Concept in mathematics
S2CID 119168700. Sinha, Dev (2010-02-20). "The homology of the little disks operad". p. 2. arXiv:math/0610236. Magnus, Wilhelm (1974). "Braid groups: A survey"
Configuration space (mathematics)
Configuration_space_(mathematics)
Theorem in category theory
ISSN 1076-9803. Loday, Jean-Louis; Vallette, Bruno (8 August 2012). Algebraic Operads. Grundlehren der mathematischen Wissenschaften. Vol. 346. Springer
Mac_Lane's_coherence_theorem
German mathematician
Connes-Kreimer's Hopf Algebra." Topology 46, 1 (2007), 39-88. Kaufmann, Ralph M. "Operads, Moduli of Surfaces and Quantum Algebras", in: N. Tongring and
Ralph_Kaufmann
Number in combinatorics
1017/S0950184300002639. Holtkamp, Ralf (2006), "On Hopf algebra structures over free operads", Advances in Mathematics, 207 (2): 544–565, arXiv:math/0407074
Schröder–Hipparchus_number
Generalization of category theory
"Categorification". arXiv:math/9802029. Leinster, Tom (2004). Higher Operads, Higher Categories. Cambridge University Press. arXiv:math.CT/0305049.
Higher_category_theory
Standard that diagrams must satisfy up to isomorphism
Différentielle Catégoriques. 41 (4): 255–304. Leinster, Tom (22 July 2004). Higher Operads, Higher Categories. Cambridge University Press. ISBN 978-0-521-53215-0
Coherency_(homotopy_theory)
American mathematician
International Congress of Mathematicians. Vol. 2. pp. 433–442. "Introduction to Operads (William Dwyer @ MSRI)". YouTube. 5 August 2014. (See operad.)
William_Gerard_Dwyer
Jones 1994, § 1 Getzler, Ezra; Jones, J. D. S. (1994-03-08). "Operads, homotopy algebra and iterated integrals for double loop spaces". arXiv:hep-th/9403055
S-object
Belgian mathematician
of Oxford in 2001, with a dissertation Ribbon Graphs and Related Operads in algebraic topology supervised by Ulrike Tillmann. After short-term positions
Nathalie_Wahl
compositions; see the introduction of Lurie's paper, leading to the notion of an operad.) Given a based space ( X , ∗ ) {\displaystyle (X,*)} , we let P ′ X = {
Path_space_fibration
American mathematician
MR 0158400 Markl, Martin; Shnider, Steve; Stasheff, James D. (2002). Operads in algebra, topology and physics. Mathematical Surveys and Monographs. Vol. 96
Jim_Stasheff
Convex polytope of parenthesizations
Sequences. OEIS Foundation. Holtkamp, Ralf (2006), "On Hopf algebra structures over free operads", Advances in Mathematics, 207 (2): 544–565, arXiv:math/0407074
Associahedron
American mathematician
1996 he received his PhD from MIT with thesis Spaces of Algebra Structures and Cohomology of Operads and advisor Michael J. Hopkins. At Northwestern University
Charles_Rezk
Higher category theory concept
Definitions of n-Category". arXiv:math/0107188. Leinster, Tom (2004a). "Operads in Higher-Dimensional Category Theory" (PDF). Theory and Applications of
Weak_n-category
Mathematical framework for knowledge representation
composition. Hypergraph Modeling language Ontology language Operad theory Orgology Universal algebra Universal logic Spivak, David I.; Kent, Robert E. (31 January
Olog
Dutch mathematician
Recently Moerdijk pursues, among other topics, research on the theory of operads, on the logic structure of quantum information theory, and on dendroidal
Ieke_Moerdijk
Russian mathematician (born 1962)
Langlands Conjecture" from algebraic curves to algebraic surfaces. In 1998 Kapranov was an Invited Speaker with talk Operads and Algebraic Geometry at the International
Mikhail_Kapranov
Mathematician
whose dissertation was entitled Symmetric Homotopy Theory for Operads and Weak Lie 3-Algebras, Du Li (2014), whose dissertation was entitled Higher Groupoid
Chenchang_Zhu
Mathematical condition
1.1. Kontsevich, Maxim; Soibelman, Yan (2000). "Deformations of algebras over operads and Deligne's conjecture". Conférence Moshé Flato 1999: Quantization
Poincaré_lemma
French mathematician (1946–2012)
1007/BF01445099. S2CID 16865683. Loday, Jean-Louis; Vallette, Bruno (2012), Algebraic Operads (PDF), Grundlehren der Mathematischen Wissenschaften, vol. 346, Berlin
Jean-Louis_Loday
Graduate-level textbooks in mathematics
has a companion volume: FTOAN/3.S Fundamentals of the Theory of Operator Algebras. Volume III, Richard V. Kadison, John R. Ringrose (1991, ISBN 978-0-8218-9469-9)
Graduate Studies in Mathematics
Graduate_Studies_in_Mathematics
American mathematician and novelist
"Grothendieck–Teichmüller groups", Grothendieck–Teichmüller Groups, Deformation and Operads, archived from the original on 2014-03-09, retrieved 2014-01-02 Schneps
Leila_Schneps
distributive laws between monads are a way to express abstractly that two algebraic structures distribute one over the other. Suppose that ( S , μ S , η S
Distributive law between monads
Distributive_law_between_monads
Alternative mathematical ordering
(1997), "From operads to 'physically' inspired theories", in Loday, Jean-Louis; Stasheff, James D.; Voronov, Alexander A. (eds.), Operads: Proceedings
Cyclic_order
French mathematician (1924–1987)
The concept of "analyseurs", reinvented by J. Peter May under the name operads. In 1958, Lazard was the first recipient of the Prix Audin, named after
Michel_Lazard
Polytope associated with combinatorial problems
(1997), "From operads to 'physically' inspired theories", in Loday, Jean-Louis; Stasheff, James D.; Voronov, Alexander A. (eds.), Operads: Proceedings
Cyclohedron
Mathematical structures in category theory
Mathematics portal Diagram (category theory) Tom Leinster (2004). Higher Operads, Higher Categories. Cambridge University Press. Bibcode:2004hohc.book.
Functor_category
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