condition. For example, a monoid object in the category of "polynomial endofunctors" on is an operad. Similarly, a symmetric operad can be defined as a monoid object in the category of -objects, where means a symmetric group. A monoid object in the category of combinatorial species is an operad in finite sets. An operad in the above sense is sometimes thought of as a generalized ring. For example, Nikolai Durov defines his generalized rings as monoid objects in the monoidal category of endofunctors on that commute with filtered colimits. This is a generalization of a ring since each ordinary ring R defines a monad that sends a set X to the underlying set of the free R-module generated by X. Understanding the axioms Associativity axiom "Associativity" means that composition of operations is associative (the function is associative), analogous to the axiom in category theory that ; it does not mean that the operations themselves are associative as operations. Compare with the associative operad, below. Associativity in operad theory means that expressions can be written involving operations without ambiguity from the omitted compositions, just as associativity for operations allows products to be written without ambiguity from the omitted parentheses. For instance, if is a binary operation, which is written as or . So that may or may not be associative. Then what is commonly written is unambiguously written operadically as . This sends to (apply on the first two, and the identity on the third), and then the on the left "multiplies" by . This is clearer when depicted as a tree: which yields a 3-ary operation: However, the expression is a priori ambiguous: it could mean , if the inner compositions are performed first, or it could mean , if the outer compositions are performed first (operations are read from right to left). Writing , this is versus . That is, the tree is missing "vertical parentheses": If the top two rows of operations are composed first (puts an upward parenthesis at the line; does the inner composition first), the following results: which then evaluates unambiguously to yield a 4-ary operation. As an annotated expression: If the bottom two rows of operations are composed first (puts a downward parenthesis at the line; does the outer composition first), following results: which then evaluates unambiguously to yield a 4-ary operation: The operad axiom of associativity is that these yield the same result,