Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16
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Abelian groups form a reflective full subcategory of groups

Statement

The full subcategory Ab of abelian groups is reflective in Grp, with reflector given by abelianisation.

Facts & Assumptions

Given: The full inclusion I:Ab↪Grp.

[L1]

Abelianisation defines a functor left adjoint to I, and every map from a group to an abelian group factors uniquely through the abelianisation quotient (Abelianisation is left adjoint to the inclusion of abelian groups).

[L2]

A full subcategory is reflective when its inclusion has a left adjoint (Reflective full subcategory and reflector).

Proof

technique · direct
1.1L1

The adjunction of [L1] exhibits the full inclusion I as a right adjoint with abelianisation as its left adjoint.

2.1step 1.1L2∎

Hence [L2] makes Ab reflective in Grp with abelianisation as reflector.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources