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Abelian groups form a reflective full subcategory of groups
Statement
The full subcategory of abelian groups is reflective in , with reflector given by abelianisation.
Facts & Assumptions
Given: The full inclusion .
Abelianisation defines a functor left adjoint to , 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).
A full subcategory is reflective when its inclusion has a left adjoint (Reflective full subcategory and reflector).
Proof
The adjunction of [L1] exhibits the full inclusion as a right adjoint with abelianisation as its left adjoint.
Hence [L2] makes reflective in with abelianisation as reflector.
Depends on
Used by
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Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 21 results over 5 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- E. Riehl, Category Theory in Context, example 4.5.13 (standard reference, not scraped)