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rational homotopy theory could be explained using Koszul duality of operads. Operads have since found many applications, such as in deformation quantization...

(where n ∈ N). In the setting of non-Σ operads (also termed nonsymmetric operads, operads without permutation), an operad A is A∞ if all of its spaces A(n)...

associative operad is self-dual, while the commutative and the Lie operad correspond to each other under this duality. Koszul duality for operads states an...

distributive over colimits. If f : O → O ′ {\displaystyle f:O\to O'} is a map of operads and, moreover, if f is a homotopy equivalence, then the ∞-category of algebras...

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...

several variables. Multicategories are also sometimes called operads, or colored operads. A (non-symmetric) multicategory consists of a collection (often...

MR 1301191 Ginzburg & Kapranov 1994, § 1.3.9. Todd Trimble, Notes on operads and the Lie operad https://ncatlab.org/nlab/show/Lie+operad v t e...

In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally...

ISBN 978-0-521-59642-8, MR 1483118 Fresse, Benoit (2017), Homotopy of Operads and Grothendieck-Teichmüller Groups: Part 2: The Applications of (Rational)...

{\mathsf {Operads}}\subset {\tfrac {1}{2}}{\mathsf {PROP}}\subset {\mathsf {PROP}}} where the first category is the category of (symmetric) operads. An important...

organizes this structure via operads, while the recent approach of Jacob Lurie deals with it using ∞ {\displaystyle \infty } -operads in ∞ {\displaystyle \infty...

graduated in 2000 with a PhD from the University of Cambridge. His PhD thesis Operads in Higher-Dimensional Category Theory was supervised by Martin Hyland....

mentoring. J. Peter May at the Mathematics Genealogy Project May, J. Peter. "Operads, Algebras, and Modules" (PDF). math.uchicago.edu. p. 2. Retrieved 28 September...

conical combinations, and convex combinations to correspond to the sub-operads where the terms sum to 1, the terms are all non-negative, or both, respectively...

geometry; supersymmetry; operads; and Hopf algebras. He retired in 2014. A 2002 book of Markl, Shnider and Stasheff Operads in algebra, topology, and...

University Press. ISBN 978-0-521-89485-2. Leinster, Tom (2004). Higher Operads, Higher Categories. London Math. Society Lecture Note Series. Vol. 298...

this may seem to be a troublesome restriction, but the payoff is that operads have certain advantages: for example, one can hybridize the concepts of...

category consisting of finite rooted trees considered as operads, whose morphisms are operad morphisms. The trees are allowed to have some edges with...

21, 1996). "The Pre-History Of Operads". In Loday, Jean-Louis; Stasheff, James D.; Voronov, Alexander A. (eds.). Operads: Proceedings of Renaissance Conferences...

algebraic surfaces. In 1998 Kapranov was an Invited Speaker with talk Operads and Algebraic Geometry at the International Congress of Mathematicians...

Kontsevich, Maxim; Soibelman, Yan (2000). "Deformations of algebras over operads and the Deligne conjecture". Conférence Moshé Flato 1999, Vol. I (Dijon)...

a structure that behaves like an algebra over a field. Algebra over an operad Alternative algebra Clifford algebra Composition algebra Differential algebra...

Furthermore, Ginzburg and Mikhail Kapranov developed Koszul duality theory for operads. In noncommutative geometry, Ginzburg defined, following earlier ideas...

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...

presheaf on the category of opetopes. higher-order operad Leinster, Tom (2004). "Opetopes". Higher Operads, Higher Categories. London Mathematical Society...

(see Mumford measure), semi-infinite cohomology, operads in quantum field theory (see Swiss-cheese operad), Deligne's and Kontsevich's conjectures on Hochschild...

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 over...

S2CID 16865683. Loday, Jean-Louis; Vallette, Bruno (2012), Algebraic Operads (PDF), Grundlehren der Mathematischen Wissenschaften, vol. 346, Berlin...

"Categorification". arXiv:math/9802029. Leinster, Tom (2004). Higher Operads, Higher Categories. Cambridge University Press. arXiv:math.CT/0305049....

algebra extension Lie algebra representation Lie bialgebra Lie coalgebra Lie operad Particle physics and representation theory Orthogonal symmetric Lie algebra...

sets and bijections between them. Trimble n.d., § 1 Trimble, Todd H. "Notes on the Lie operad" (PDF). University of Chicago. Retrieved 2022-09-27. v t e...

Kontsevich, Maxim; Soibelman, Yan (2000). "Deformations of algebras over operads and Deligne's conjecture". Conférence Moshé Flato 1999: Quantization, Deformations...

associahedron (also called the Stasheff polytope), ideas leading to the theory of operads; homotopy theoretic approaches to Hilbert's fifth problem on the characterization...

PhD from MIT with thesis Spaces of Algebra Structures and Cohomology of Operads and advisor Michael J. Hopkins. At Northwestern University Rezk was a faculty...

195–213. Ginzburg, Victor; Kapranov, Mikhail (1994). "Koszul duality for operads". Duke Mathematical Journal. 76: 203–273. arXiv:0709.1228. doi:10...

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"...

Foundation. Holtkamp, Ralf (2006), "On Hopf algebra structures over free operads", Advances in Mathematics, 207 (2): 544–565, arXiv:math/0407074, doi:10...

International Congress of Mathematicians. Vol. 2. pp. 433–442. "Introduction to Operads (William Dwyer @ MSRI)". YouTube. 5 August 2014. (See operad.)...

1017/S0950184300002639. Holtkamp, Ralf (2006), "On Hopf algebra structures over free operads", Advances in Mathematics, 207 (2): 544–565, arXiv:math/0407074, doi:10...

S2CID 121944962. Ginzburg, V.; Kapranov, M. (1994). "Koszul duality for operads". Duke Math. J. 76: 203–273. arXiv:0709.1228. doi:10.1215/s0012-7094-94-07608-4...

Deformation Theory. I (PDF) Loday, Jean-Louis; Vallette, Bruno (2012), Algebraic Operads, Grundlehren der mathematischen Wissenschaften, vol. 346, Heidelberg, New...

to non-mathematicians." Spivak has also studied dynamical systems and operads, originating the operadic approach to wiring diagram syntax. His unpublished...

of San Diego Williams College Thesis Tesselations of moduli spaces and the mosaic operad (1999) Doctoral advisor Jack Morava Website satyandevadoss.org...

(1997), "From operads to 'physically' inspired theories", in Loday, Jean-Louis; Stasheff, James D.; Voronov, Alexander A. (eds.), Operads: Proceedings...

of a cactus operad on the free loop space L M {\displaystyle LM} . The cactus operad is weakly equivalent to the framed little disks operad and its action...

Getzler & 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...

Kontsevich, Maxim; Soibelman, Yan (2000). "Deformations of algebras over operads and Deligne's conjecture". Conférence Moshé Flato 1999: Quantization, Deformations...

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 with prototypical algebras....

ISSN 1076-9803. Loday, Jean-Louis; Vallette, Bruno (8 August 2012). Algebraic Operads. Grundlehren der mathematischen Wissenschaften. Vol. 346. Springer. doi:10...

165-195. Kaufmann, Ralph M.; Livernet, Muriel and Penner, Robert C. "Arc Operads and Arc Algebras". Geometry and Topology 7 (2003), 511-568. Kaufmann, Ralph...

Borcherds's later work on singular commutative rings, algebras over certain operads on curves introduced by Huang, Kriz, and others, D-module-theoretic objects...

Mathematical Social Sciences, 114 (2021), 39–57 "Formality of the little N-discs operad", with P. Lambrechts, Memoirs of the American Mathematical Society, 239...

only finitely many cells in each dimension. operad The portmanteau of “operations” and “monad”. See operad. orbibundle orbibundle. orbit category orientation...

Eilenberg, Homological algebra Fresse, Benoit (21 April 2017). Homotopy of Operads and Grothendieck-Teichmuller Groups. Mathematical Surveys and Monographs...

(1997), "From operads to 'physically' inspired theories", in Loday, Jean-Louis; Stasheff, James D.; Voronov, Alexander A. (eds.), Operads: Proceedings...

Definitions of n-Category". arXiv:math/0107188. Leinster, Tom (2004a). "Operads in Higher-Dimensional Category Theory" (PDF). Theory and Applications of...

equivalent 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...

Sistemy, 9: 30–43, MR 0209178. Weber, Mark (2013), "Free products of higher operad algebras", Theory and Applications of Categories, 28 (2): 24–65. Weisstein...

{\displaystyle k[x]} . Loday, Jean-Louis; Vallette, Bruno (2012). Algebraic operads. Grundlehren der Mathematischen Wissenschaften. Vol. 346. Berlin: Springer-Verlag...

Hecke algebras. For category algebras. For small categories. For shuffle operads. The lemma has been used to prove the Poincaré–Birkhoff–Witt theorem. Rogalski...

geometry". arXiv:2109.14594 [math.AG]. Spectral algebraic geometry - Rezk Operads and Sheaf Cohomology - JP May - E ∞ {\displaystyle E_{\infty }} -rings...

Frabetti, Alessandra; Goichot, François (eds.). Dialgebras and Related Operads. Lecture Notes in Mathematics. Vol. 1763. Springer. pp. 7–66. doi:10.1007/3-540-45328-8_2...

Différentielle Catégoriques. 41 (4): 255–304. Leinster, Tom (22 July 2004). Higher Operads, Higher Categories. Cambridge University Press. ISBN 978-0-521-53215-0...

S2CID 50187913. Maxim Kontsevich and Yan Soibelman. “Deformations of algebras over operads and the Deligne conjecture”. In: Conférence Moshé Flato 1999, Vol. I (Dijon)...

Age. April 30, 1955. p. 19. "Strauss Created New Orchestra in Writing His Operad Ariadne". Montreal Gazette. November 26, 1946. Rebecca Paller (October 12...

and musical composition. Hypergraph Modeling language Ontology language Operad theory Orgology Universal algebra Universal logic Spivak, David I.; Kent...

 337–350, MR 1767430. Loday, Jean-Louis; Vallette, Bruno (2012), Algebraic operads (PDF), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles...

(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...

discussed in that article. See also Lie operad for the use of a free Lie algebra in the construction of the operad. Free object Free algebra Free group Berstel...

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...

Mathematics portal Diagram (category theory) Tom Leinster (2004). Higher Operads, Higher Categories. Cambridge University Press. Bibcode:2004hohc.book....

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...

Recently Moerdijk pursues, among other topics, research on the theory of operads, on the logic structure of quantum information theory, and on dendroidal...

University of Oxford in 2001, with a dissertation Ribbon Graphs and Related Operads in algebraic topology supervised by Ulrike Tillmann. After short-term positions...

World Scientific. ISBN 9789814520430. Kontsevich, Maxim (1 April 1999). "Operads and Motives in Deformation Quantization". Letters in Mathematical Physics...

F.; Markl, M. (1997). "Distributive laws, bialgebras, and cohomology". Operads: Proceedings of Renaissance Conferences. Contemporary Mathematics. Vol...

diagrammatic formalism of algebras which is described in Algebra+Homotopy=Operad explaining how to visually think about this higher homotopies. This intuition...

correspondence between Lawvere theories and abstract clones. Term algebra Operad Denecke, Klaus (2003). "Menger algebras and clones of terms". East–West...

Chapoton, F.; Livernet, M. (2001), "Pre-Lie algebras and the rooted trees operad", International Mathematics Research Notices, 2001 (8): 395–408, doi:10...

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 = {...

"Grothendieck–Teichmüller groups", Grothendieck–Teichmüller Groups, Deformation and Operads, archived from the original on 2014-03-09, retrieved 2014-01-02 Schneps...

of algebra for T, that satisfies the laws up to coherent isomorphisms. Operad Shulman, Michael A. (2012). "Not every pseudoalgebra is equivalent to a...

to: E-infinity ring, a commutative ring construct in homotopy theory E∞-operad, a set of contractible maps that are commutative and associative E-infinity...

Press. arXiv:1612.09375. Bibcode:2016arXiv161209375L. Leinster, Higher Operads, Higher Categories Archived 2024-07-13 at the Wayback Machine, 2003. Emily...

Theory and Applications András Vasy 2015 978-1-4704-1881-6 170 Colored Operads Donald Yau 2016 978-1-4704-2723-8 171 Nonlinear Elliptic Equations of the...

16–2013, art. 13.1.4 Dzhumadildaev A.S., Zusmanovich P., The alternative operad is not Koszul // Experimental Mathematics. – 2011. – V. 20, No.2. – P. 138–144...

(2020), whose dissertation was entitled Symmetric Homotopy Theory for Operads and Weak Lie 3-Algebras, Du Li (2014), whose dissertation was entitled...