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reflective subcategory
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(Definition)
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Let $\mathcal{C}$ be a category and $\mathcal{D}$ a subcategory of $\mathcal{C}$ $\mathcal{D}$ is called a reflective subcategory of $\mathcal{C}$ if the inclusion functor $\operatorname{Inc}:\mathcal{D}\to \mathcal{C}$ has a left adjoint. More explicitly, $\mathcal{D}$ in $\mathcal{C}$ is reflective iff for every object $A$ in $\mathcal{C}$ there is an object $B$ in $\mathcal{D}$ and a morphism $f:A\to B$ such that any morphism $g:A\to C$ can be uniquely factored through $f$ that is, there is a unique morphism $h:B\to C$ such that $g=h\circ f$
The left adjoint is called the reflection functor and the mapped objects and morphisms are called the reflections (of the objects and morphisms being mapped by the reflection functor).
Some of the most common reflective subcategories are
Remark. If the inclusion functor has a right adjoint, then the subcategory is said to be coreflective. In other words, $\mathcal{D}$ in $\mathcal{C}$ is coreflective iff for any object $A\in \mathcal{C}$ there is an object $B\in \mathcal{D}$ and a morphism $f:B\to A$ such that any morphism $g:C\to A$ can be uniquely factored through $f$ (by a unique morphism $f:C\to B$ . For example, the subcategory of torsion abelian groups in the category of abelian groups is coreflective. The coreflection of an abelian group is its torsion
subgroup.
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"reflective subcategory" is owned by CWoo.
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(view preamble | get metadata)
| Also defines: |
reflection functor, reflection, coreflective, coreflection |
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Cross-references: torsion subgroup, torsion, lattice of ideals, lattice, complete, field of fractions, integral domains, fields, functor, abelianization, groups, abelian groups, morphism, object, iff, left adjoint, inclusion functor, subcategory, category
There are 14 references to this entry.
This is version 1 of reflective subcategory, born on 2007-06-04.
Object id is 9524, canonical name is ReflectiveSubcategory.
Accessed 2616 times total.
Classification:
| AMS MSC: | 18A40 (Category theory; homological algebra :: General theory of categories and functors :: Adjoint functors ) |
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Pending Errata and Addenda
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