symmetric monoidal (∞,1)-category of spectra
monoid theory in algebra:
The free commutative monoid on a set is the commutative monoid whose elements are formal -linear combinations of elements of .
Let
be the forgetful functor from the category CMon of commutative monoids, to the category Set of sets. This has a left adjoint free construction:
This is the free commutative monoid functor. For Set, the free commutative monoid CMon is the free object on with respect to this free-forgetful adjunction.
Of course, this notion is meant to be invariant under isomorphism: it doesn’t depend on the left adjoint chosen. Thus, if a functor of the form is representable by a commutative monoid , then we may say is a free commutative monoid on . A specific choice of isomorphism
corresponds, via the Yoneda lemma, to a function which exhibits , or rather its image under this function, as a specific basis of . If is so equipped with such a universal arrow , then it is harmless to call “the” free commutative monoid on .
Explicit descriptions of free commutative monoid are discussed below.
A formal linear combination of elements of a set is a function
such that only finitely many of the values are non-zero.
Identifying an element with the function which sends to and all other elements to 0, this is written as
In this expression one calls the coefficient of in the formal linear combination.
Definition of formal linear combinations makes sense with coefficients in any commutative monoid , not necessarily the natural numbers.
For Set, the monoid of formal linear combinations is the monoid whose underlying set is that of formal linear combinations, def. , and whose monoid operation is the pointwise addition in :
The free commutative monoid on is, up to isomorphism, the monoid of formal linear combinations, def. , on .
For a set, the free commutative monoid is the biproduct in CMon of -copies of with itself:
For a rig and a set, the tensor product of commutative monoids is the free module over on the basis .
For a rig, the tensor product of commutative monoids is the commutative monoid underlying the rig of polynomials over .
Last revised on May 21, 2021 at 22:29:48. See the history of this page for a list of all contributions to it.