nLab localization of a monoid

Idea

The concept of localization of a monoid, the localization of a category with a single object. Could be generalized to localization of monoid objects in a cartesian monoidal category $C$. The localization of a ring is a localization of a monoid object in Ab.

Definition

For commutative monoids

Let $(M,1,*)$ be a commutative monoid object in $Set$ and let $S$ be a commutative submonoid of $M$. Then the localization of $M$ away from $S$, $S^{-1}M$, is the set of equivalences on $M \times S$, $(m_1,s_1) \sim (m_2,s_2)$ such that there is an element $u:S$ where $m_1 \cdot s_2 \cdot u = m_2 \cdot s_1 \cdot u$.

Write $m s^{-1}$ for the equivalence class of $(m,s)$. On this set, the product is defined by

$$(m_1 s_1^{-1}) \cdot (m_2 s_2^{-1}) \coloneqq (m_1 \cdot m_2) (s_1 \cdot s_2)^{-1} \,.$$

For non-commutative monoids

Analogue of noncommutative localization for noncommutative monoid objects in $Set$ instead of $Ab$.

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Group completion

The group completion of a commutative monoid $M$ is the localization of $M$ away from $M$.

Related concepts

• localization

• localization of a ring