nLab free commutative monoid

1. Idea

The free commutative monoid $\mathbb{N}[S]$ on a set $S$ is the commutative monoid whose elements are formal $\mathbb{N}$-linear combinations of elements of $S$.

2. Definition

Definition 2.1. Let

$$U \colon CMon \longrightarrow Set$$

be the forgetful functor from the category CMon of commutative monoids, to the category Set of sets. This has a left adjoint free construction:

$$\mathbb{N}[-] \colon Set \longrightarrow CMon \,.$$

This is the free commutative monoid functor. For $S \in$ Set, the free commutative monoid $\mathbb{N}[S] \in$ CMon is the free object on $S$ 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 $\hom_{Set}(S, U-): CMon \to Set$ is representable by a commutative monoid $M$, then we may say $M$ is a free commutative monoid on $S$. A specific choice of isomorphism

$$\hom_{CMon}(M, -) \cong \hom_{Set}(S, U-)$$

corresponds, via the Yoneda lemma, to a function $S \to U M$ which exhibits $S$, or rather its image under this function, as a specific basis of $M$. If $M$ is so equipped with such a universal arrow $S \to U M$, then it is harmless to call $M$ “the” free commutative monoid on $S$.

Explicit descriptions of free commutative monoid are discussed below.

3. Properties

In terms of formal linear combinations

Definition 3.1. A formal linear combination of elements of a set $S$ is a function

$$a : S \to \mathbb{N}$$

such that only finitely many of the values $a_s \in \mathbb{N}$ are non-zero.

Identifying an element $s \in S$ with the function $S \to \mathbb{N}$ which sends $s$ to $1 \in \mathbb{N}$ and all other elements to 0, this is written as

$$a = \sum_{s \in S} a_s \cdot s \,.$$

In this expression one calls $a_s \in \mathbb{N}$ the coefficient of $s$ in the formal linear combination.

Remark 3.2. Definition 3.1 of formal linear combinations makes sense with coefficients in any commutative monoid $M$, not necessarily the natural numbers.

$$M[S] \coloneqq \mathbb{N}[S] \otimes M \,.$$

Definition 3.3. For $S \in$ Set, the monoid of formal linear combinations $\mathbb{N}[S]$ is the monoid whose underlying set is that of formal linear combinations, def. 3.1, and whose monoid operation is the pointwise addition in $\mathbb{N}$:

$$(\sum_{s \in S} a_s \cdot s) + (\sum_{s \in S} b_s \cdot s) = \sum_{s \in S} (a_s + b_s) \cdot s \,.$$

Proposition 3.4. The free commutative monoid on $S \in Set$ is, up to isomorphism, the monoid of formal linear combinations, def. 3.3, on $S$.

Proposition 3.5. For $S$ a set, the free commutative monoid $\mathbb{N}[S]$ is the biproduct in CMon of ${|S|}$-copies of $\mathbb{N}$ with itself:

$$\mathbb{N}[S] \simeq \oplus_{s \in S} \mathbb{N} \,.$$

4. Examples

• For $R$ a rig and $S$ a set, the tensor product of commutative monoids $\mathbb{N}[S] \otimes R$ is the free module over $R$ on the basis $S$.

• For $R$ a rig, the tensor product of commutative monoids $\mathbb{N}[\mathbb{N}]\otimes R$ is the commutative monoid underlying the rig of polynomials over $R$.

5. Related concepts

• free monoid

• free abelian group