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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- Commons category
- Operads
- Stack Exchange tag
- math.stackexchange.com/tags/operads
- time of discovery or invention
- 1969-00-00
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Article
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.