, a composition function satisfying the following coherence axioms: identity: associativity: Symmetric operad A symmetric operad (often just called operad) is a non-symmetric operad as above, together with a right action of the symmetric group on for , denoted by and satisfying equivariance: given a permutation , (where on the right hand side refers to the element of that acts on the set by breaking it into blocks, the first of size , the second of size , through the th block of size , and then permutes these blocks by , keeping each block intact) and given permutations , (where denotes the element of that permutes the first of these blocks by , the second by , etc., and keeps their overall order intact). The permutation actions in this definition are vital to most applications, including the original application to loop spaces. Morphisms A morphism of operads consists of a sequence that: preserves the identity: preserves composition: for every n-ary operation and operations , preserves the permutation actions: . Operads therefore form a category denoted by . In other categories So far operads have only been considered in the category of sets. More generally, it is possible to define operads in any symmetric monoidal category C . In that case, each is an object of C, the composition is a morphism in C (where denotes the tensor product of the monoidal category), and the actions of the symmetric group elements are given by isomorphisms in C. A common example is the category of topological spaces and continuous maps, with the monoidal product given by the cartesian product. In this case, a topological operad is given by a sequence of spaces (instead of sets) . The structure maps of the operad (the composition and the actions of the symmetric groups) are then assumed to be continuous. The result is called a topological operad. Similarly, in the definition of a morphism of operads, it would be necessary to assume that the maps involved are continuous. Other common settings to define operads include, for example, modules over a commutative ring, chain complexes, groupoids (or even the category of categories itself), coalgebras, etc. Algebraist definition Given a commutative ring R we consider the category of modules over R. An operad over R can be defined as a monoid object in the monoidal category of endofunctors on (it is a monad) satisfying some finiteness