Reflective sub-category

In the mathematical branch of category theory, a reflective sub-category is a sub-category with an additional property. The objects of the subcategory arise from the objects of the upper class through a functorial process that can be imagined as a kind of completion.

definition

Let it be a sub-category of . is called reflective (in ) if the inclusion functor is right adjoint . A functor that is too left adjoint is called a reflector . ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle I\colon {\mathcal {D}}\rightarrow {\mathcal {C}}}$ ${\displaystyle I}$ ${\displaystyle {\mathcal {C}}\rightarrow {\mathcal {D}}}$

A sub-category of koreflective (in ) is called dual to this if the inclusion function is left adjoint . A functor that is adjoint to the right is called a core reflector . ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle I\colon {\mathcal {D}}\rightarrow {\mathcal {C}}}$ ${\displaystyle I}$ ${\displaystyle {\mathcal {C}}\rightarrow {\mathcal {D}}}$

Examples

• Let be the category of the metric spaces with the Lipschitz continuous functions with Lipschitz constant as morphisms. Then the sub-category of full metric spaces is reflective. The functor that assigns the completion defined by means of equivalence classes of Cauchy sequences to every metric space is a reflector. Other methods of completion lead to different reflectors, but every two such reflectors are of course equivalent , see below. ${\displaystyle {\mathcal {M}}}$ ${\displaystyle \leq 1}$ ${\displaystyle {\overline {\mathcal {M}}}}$ ${\displaystyle {\mathcal {M}}\rightarrow {\overline {\mathcal {M}}}}$

• Let be the category of integrity rings with the injective , one-preserving ring homomorphisms. Then the sub-category of bodies is a reflective sub-category of . A reflector is the functor that assigns its quotient field (as a set of equivalence classes of pairs of the ring) to each integrity ring. In this example, the reflector completes the integrity ring around the missing inverse of a body. ${\displaystyle {\mathcal {Int}}}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle {\mathcal {Int}}}$ ${\displaystyle {\mathcal {Int}}\rightarrow {\mathcal {K}}}$

• The category of compact Hausdorff spaces is a reflective sub-category of the category of all topological spaces . The Stone-Čech compacting is a reflector.

• The category of compact groups is a reflective sub-category of the category of all topological groups . The Bohr compactification is a reflector.

• Counterexample: The algebraic closure of a body is not a reflector, more precisely: The sub-category of algebraically closed bodies is not reflective in the category of bodies.

• The category of Abelian torsion groups is co-reflective (but not reflective) in the category of Abelian groups. The formation of the torsion subgroup is a core reflector.

• The category of locally connected spaces is a core-reflective sub-category of the category of all topological spaces. A core reflector is the transition to the coarsest locally connected topology, which is finer than the given one.

properties

• With the exception of natural isomorphism, reflectors and core reflectors are clearly defined. This is simply because the left or right adjoint of a functor, if it exists, is uniquely determined except for natural isomorphism.

• In order to be able to understand reflectors as completions, the completion, i.e. the application of the reflector, should not bring anything new to an object of the reflective subcategory. With the additional requirement that the subcategory is full, that is, that the inclusion functor is a full functor , this actually applies:

It is a full and reflective sub-category of reflector . Then the restriction from to is naturally isomorphic to the identical functor . ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle R\colon {\mathcal {C}}\rightarrow {\mathcal {D}}}$ ${\displaystyle R}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle \mathrm {id} _{\mathcal {D}}}$

The requirement of completeness cannot be dispensed with here. Therefore, in M. Brandenburg's textbook, for example, completeness is already built into the definition of the reflective sub-category.

• If it is a full and reflective subcategory of and is complete , then is also complete. ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {D}}}$

The following applies in dual terms : If it is a full and core-reflective sub-category of and is complete , then is also complete. ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {D}}}$

Since the category of topological spaces is complete and the category of compact Hausdorff spaces according to the above example is full and reflective , the category of compact Hausdorff spaces is also completely complete because of this property. Note, however, that a Kolimes formed in ( e.g. a co-product ) of compact Hausdorff spaces generally does not correspond to the one formed in. ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {K}}}$

Individual evidence

1. ↑ Horst Herrlich, George E. Strecker: Category Theory , Allyn and Bacon Inc. 1973, Chapter X: Reflective Subcategories , Definition 36.1 (2)
2. ↑ Bodo Pareigis: Categories and Functors , Teubner-Verlag 1969, Chapter 2.4: Reflexive sub-categories
3. ↑ Saunders Mac Lane: Categories for the Working Mathematician , Springer-Verlag 1997, ISBN 0-387-98403-8 , Chapter IV.3: Reflective Subcategories
4. ↑ Martin Brandenburg: Introduction to Category Theory , Springer-Verlag 2017, ISBN 978-3-662-53520-2 , example 7.6.5
5. ↑ Martin Brandenburg: Introduction to Category Theory , Springer-Verlag 2017, ISBN 978-3-662-53520-2 , example 7.6.8
6. ↑ Horst Herrlich, George E. Strecker: Category Theory , Allyn and Bacon Inc. 1973, Chapter X, Example 36.2 (3)
7. ↑ Horst Herrlich, George E. Strecker: Category Theory , Allyn and Bacon Inc. 1973, Chapter X, Example 36.2 (2)
8. ↑ Martin Brandenburg: Introduction to Category Theory , Springer-Verlag 2017, ISBN 978-3-662-53520-2 , example 7.6.3
9. ↑ Bodo Pareigis: Categories and Functors , Teubner-Verlag 1969, Lemma in Chapter 2.4
10. ↑ Horst Herrlich, George E. Strecker: Category Theory , Allyn and Bacon Inc. 1973, Chapter X, Paragraph 36
11. ↑ Martin Brandenburg: Introduction to Category Theory , Springer-Verlag 2017, ISBN 978-3-662-53520-2 , Chapter 7.6: Reflective subcategories
12. ↑ Martin Brandenburg: Introduction to Category Theory , Springer-Verlag 2017, ISBN 978-3-662-53520-2 , sentence 7.6.7