An operad is binary if it is generated by elements of arity 2.
An operad is ternary if it is generated by elements of arity 3.
An operad is quadratic if it has a presentation in which every relation involves only compositions of 2 generators.
An operad is cubic if it has a presentation in which every relation involves only compositions of 3 generators.
An operad is Koszul if it is binary quadratic and further satisfies an homological criterion.
An operad is free if it is isomorphic to the free operad on some generators.
An operad is monomial if it has a presentation in which every relation is a monomial in the generators.
An operad is Hopf if it the space of operations of every given arity admits a coassociative and cocommutative coproduct, which is compatible with the compositions. This means that the operad can be considered as an operad in a monoidal category of coalgebras.
An operad is cyclic if the operations of arity n with one output can be considered as operations with n+1 inputs, in such a way that Id is invariant by exchange of its 2 inputs. For symmetric operads, this means that there is an action of the bigger symmetric group.
An operad is anticyclic if the operations of arity n with one output can be considered as operations with n+1 inputs, in such a way that Id is anti-invariant by exchange of its 2 inputs. For symmetric operads, this means that there is an action of the bigger symmetric group.
An operad has the Nielsen-Schreier property if every sub-algebra of a free algebra is free.
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Operadia