hyperplane, the infinite hyper-octant, and the infinite simplex. This formalizes what is meant by being or the standard simplex being model spaces, and such observations as that every bounded convex polytope is the image of a simplex. Here suboperads correspond to more restricted operations and thus more general theories. Commutative-ring operad and Lie operad The commutative-ring operad is an operad whose algebras are the commutative rings. It is defined by , with the obvious action of and operadic composition given by substituting polynomials (with renumbered variables) for variables. A similar operad can be defined whose algebras are the associative, commutative algebras over some fixed base field. The Koszul-dual of this operad is the Lie operad (whose algebras are the Lie algebras), and vice versa. Free Operads Typical algebraic constructions (e.g., free algebra construction) can be extended to operads. Let denote the category whose objects are sets on which the group acts. Then there is a forgetful functor , which simply forgets the operadic composition. It is possible to construct a left adjoint to this forgetful functor (this is the usual definition of free functor). Given a collection of operations E, is the free operad on E. Like a group or a ring, the free construction allows to express an operad in terms of generators and relations. By a free representation of an operad , we mean writing as a quotient of a free operad where E describes generators of and the kernel of the epimorphism describes the relations. A (symmetric) operad is called quadratic if it has a free presentation such that is the generator and the relation is contained in . Operads in homotopy theory In , Stasheff writes: Operads are particularly important and useful in categories with a good notion of "homotopy", where they play a key role in organizing hierarchies of higher homotopies. See also PRO (category theory) Algebra over an operad Higher-order operad E∞-operad Pseudoalgebra Multicategory Notes Citations References Miguel A. Mendéz (2015). Set Operads in Combinatorics and Computer Science. SpringerBriefs in Mathematics. . Samuele Giraudo (2018). Nonsymmetric Operads in Combinatorics. Springer International Publishing. . External links https://golem.ph.utexas.edu/category/2011/05/an_operadic_introduction_to_en.html Abstract algebra Category theory