localization of a monoid in nLab

Contents

Contents

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 CC. The localization of a ring is a localization of a monoid object in Ab.

Definition

For commutative monoids

Let (M,1,*)(M,1,*) be a commutative monoid object in SetSet and let SS be a commutative submonoid of MM. Then the localization of MM away from SS, S −1MS^{-1}M, is the set of equivalences on M×SM \times S, (m 1,s 1)∼(m 2,s 2)(m_1,s_1) \sim (m_2,s_2) such that there is an element u:Su:S where m 1⋅s 2⋅u=m 2⋅s 1⋅um_1 \cdot s_2 \cdot u = m_2 \cdot s_1 \cdot u.

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

(m 1s 1 −1)⋅(m 2s 2 −1)≔(m 1⋅m 2)(s 1⋅s 2) −1. (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 SetSet instead of AbAb.

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Group completion

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

• localization

• localization of a ring