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Reflective Subcategories and the Adjoint Functor Theorems

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1 · Prerequisites
Abstract Algebra
Category Theory
Combinatorics
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Definition

Reflective full subcategory and reflector

Definition

Coreflective full subcategory and coreflector

Theorem

A full subcategory is reflectively structured exactly when universal arrows are supplied at every ambient object

Theorem

The counit of a reflection is an isomorphism

Theorem

An ambient object lies in the essential image of a reflective inclusion exactly when its reflection unit is invertible

Theorem

A reflective inclusion creates every ambient limit in the ordinary isomorphism-invariant sense

Theorem

A reflective subcategory has every ambient colimit, obtained by reflecting an ambient colimit

Corollary

A reflective subcategory of a complete category is complete

Definition

Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms

Theorem

Mutual factorisation is an equivalence relation on monomorphisms into an object and dually on epimorphisms out of it

Theorem

Subobjects and quotient objects form oppositely oriented partially ordered collections

Definition

Intersection of a supplied family of subobjects as its greatest lower bound

Lemma

Wide pullbacks compute intersections of supplied set-indexed subobject representatives independently of the representatives

Definition

Well-powered and co-well-powered categories, and supplied well-powerings

Definition

Separating and coseparating sets of objects

Theorem

In a locally small category, separating and coseparating sets are equivalently jointly faithful families of representables

Definition

Weakly initial object and jointly weakly initial set

Theorem

A complete locally small category with a jointly weakly initial set has an initial object, without class-indexed choice

Definition

The solution-set condition for a functor, stated object by object

Theorem

The solution-set condition at an object is exactly a jointly weakly initial set in its comma category

Lemma

A comma-category projection strictly creates the limits preserved by the functor

Theorem

General adjoint functor theorem, objectwise initial-object form

Theorem

General adjoint functor theorem, data-supplied functor form

Theorem

A complete locally small category with a small coseparating set and intersections of all subobject collections has an initial object

Theorem

Special adjoint functor theorem, objectwise form with explicit intersection smallness or preservation data

Theorem

Special adjoint functor theorem, data-supplied functor form

Remark

Choice and smallness ledger for the initial-object lemma, GAFT, and SAFT

Corollary

A category satisfying the explicit SAFT intersection hypotheses is cocomplete

Corollary

With the objectwise SAFT universal arrows supplied, a continuous Set-valued functor from a chosen-well-powered SAFT category is representable

Theorem

Freyd's representability theorem for continuous Set-valued functors satisfying a solution set condition

Remark

Why completeness alone cannot replace a solution set or the SAFT smallness hypotheses

Theorem

Under dependent choice, the unit interval is a coseparating object in compact Hausdorff spaces

Proposition

Under the ultrafilter lemma and dependent choice, compact Hausdorff spaces satisfy the explicit SAFT hypotheses for their inclusion into topological spaces

Theorem

With the SAFT initial comma objects supplied for all spaces, they assemble into the compact-Hausdorff reflection and agree on Tychonoff spaces with the constructed Stone-Cech adjunction

Theorem

Normal-subgroup quotients of a fixed free group give a canonical solution set for the underlying-set functor on groups

Corollary

GAFT recovers the published free-group adjunction, and the comma-initial criterion the abelianisation adjunction

Theorem

With the ultrafilter lemma, dependent choice, and a supplied family of SAFT initial objects, compact Hausdorff spaces form a reflective full subcategory of topological spaces

Theorem

Abelian groups form a reflective full subcategory of groups

Theorem

Commutative rings form a reflective full subcategory of rings

Theorem

Torsion-free abelian groups form a reflective full subcategory of abelian groups

False statement

FALSE: A continuous functor on a complete category necessarily has a left adjoint

False statement

FALSE: Every reflective subcategory is closed under ambient colimits

False statement

FALSE: A reflective inclusion creates colimits

False statement

FALSE: A subobject is a monomorphism rather than an equivalence class of representatives

Sources