In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad , one defines an algebra over to be a set together with concrete operations on this set which behave just like the abstract operations of . For instance, there is a Lie operad such that the algebras over are precisely the Lie algebras; in a sense abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group is to its group representations.
In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output...
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 replacing...
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)...
More abstractly, in the language of operad theory, one can consider vector spaces to be algebras over the operad R ∞ {\displaystyle \mathbf {R} ^{\infty...
introduced the notion of a quadratic operad and defined the quadratic dual of such an operad. Very roughly, an operad is an algebraic structure consisting...
algebra over the little interval operad. This is an example of an A ∞ {\displaystyle A_{\infty }} -operad, i.e. an operad of topological spaces which is...
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally...
several variables. Multicategories are also sometimes called operads, or colored operads. A (non-symmetric) multicategory consists of a collection (often...
finite products). A more recent development in category theory is operad theory – an operad is a set of operations, similar to a universal algebra, but restricted...
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...
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...
a structure that behaves like an algebra over a field. Algebra over an operad Alternative algebra Clifford algebra Differential algebra Free algebra Geometric...
of San Diego Williams College Thesis Tesselations of moduli spaces and the mosaic operad (1999) Doctoral advisor Jack Morava Website satyandevadoss.org...
and musical composition. Hypergraph Modeling language Ontology language Operad theory Orgology Universal algebra Universal logic Spivak, David I.; Kent...
Press. ISBN 978-0-521-89485-2. Leinster, Tom [in German] (2004). Higher Operads, Higher Categories. London Math. Society Lecture Note Series. Vol. 298...
geometry; supersymmetry; operads; and Hopf algebras. He retired in 2014. A 2002 book of Markl, Shnider and Stasheff Operads in algebra, topology, and...
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...
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...
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