symmetric monoidal (∞,1)-category of spectra
Let be a commutative ring.
(so that we ignore coefficients of zero).
which on monomials is given by
Moreover, there is a noncommutative analogue of polynomial ring on a set , efficiently described as the free -module generated by the (underlying set of the) free monoid on . This carries also a ring structure, with ring multiplication induced from the monoid multiplication. A far-reaching generalization of this construction is given at distributive law.
Finally: polynomial algebras may be regarded as graded algebras (graded over ). Specifically: let us regard as the free -module generated by (the underlying set of) the free commutative monoid . The monoid homomorphism induced by the unique function gives an -fibering of over , with typical fiber whose elements are called monomials of degree . Then the homogeneous component of degree in is the -submodule generated by the subset . The elements of this component are called homogeneous polynomials of degree .
from the singleton to the set underlying . Take . Using -linearity, this is directly seen to yield the desired bijection.
() that take a tuple of elements to . Formally, it takes this tuple to the value of under the unique algebra map that extends the mapping . Here the are appropriate coproduct inclusions (in the category of commutative rings), where . A particularly important case of substitution is the case and , where the map is ordinary substitution . This is a special case of the more general notion of Tall-Wraith monoid.