operad
Sign in to saveIn 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 O, one defines an algebra over O to be a set together with concrete operations on this set that behave just like the abstract operations of O. For instance, there is a Lie operad L such that the algebras over L are precisely the Lie algebras; in a sense L abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group
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Encyclopedic overview
31 sectionsContents
- History
- Intuition
- Definition
- Non-symmetric operad
- Symmetric operad
- Morphisms
- In other categories
- Algebraist definition
- Understanding the axioms
- Associativity axiom
- Identity axiom
- Examples
- Endomorphism operad in sets and operad algebras
- Endomorphism operad in vector spaces and operad algebras
- "Little something" operads
- Rooted trees
- Swiss-cheese operad
- Associative operad
- Terminal symmetric operad
- Operads from the braid groups
- Linear algebra
- Commutative-ring operad and Lie operad
- Free operads
- Clones
- Operads in homotopy theory
- Higher-order operad
- See also
- Notes
- Citations
- References
- External links
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 O, one defines an algebra over O to be a set together with concrete operations on this set that behave just like the abstract operations of O. For instance, there is a Lie operad L such that the algebras over L are precisely the Lie algebras; in a sense L abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group is to its group actions.
== History == Operads originate in algebraic topology; they were introduced to characterize iterated loop spaces by J. Michael Boardman and Rainer M. Vogt in 1968 and by J. Peter May in 1972.
Excerpted from Wikipedia’s “operad” article, available under the CC BY-SA 4.0 licence.