Generator and co-generator

Generator and cogenerator are terms from the mathematical branch of category theory . They are objects of a category that have a certain relationship to any objects in the category.

Definitions

A lot of objects of a category is a set of generators for when it comes to two different morphisms a and a morphism are with . ${\displaystyle \{G_{i}\mid i\in I\}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle f,g:A\rightarrow B}$ ${\displaystyle i\in I}$ ${\displaystyle h:G_{i}\rightarrow A}$ ${\displaystyle f\circ h\not =g\circ h}$

An object from is called a generator for if the one-element set is a generator for . ${\displaystyle G}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \{G\}}$ ${\displaystyle {\mathcal {C}}}$

The term cogenerator is dual to this:

A lot of objects of a class is a lot of cogenerators for when it comes to two different morphisms a and a morphism are with . ${\displaystyle \{K_{i}\mid i\in I\}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle f,g:A\rightarrow B}$ ${\displaystyle i\in I}$ ${\displaystyle h:B\rightarrow K_{i}}$ ${\displaystyle h\circ f\not =h\circ g}$

An object from is called a cogenerator for if the one-element set is a cogenerator for . ${\displaystyle K}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \{K\}}$ ${\displaystyle {\mathcal {C}}}$

Instead generator and cogenerator can also find the name separator and Koseparator .

Examples

• In the category of sets, every non-empty set is a generator, because if there are different mappings, for example , then the map , which is constantly the same , performs what is required. ${\displaystyle {\mathcal {Set}}}$ ${\displaystyle G}$ ${\displaystyle f,g:A\rightarrow B}$ ${\displaystyle f(a)\not =g(a)}$ ${\displaystyle h:G\rightarrow A}$ ${\displaystyle a}$

Any set with at least two elements is a cogenerator in , because if there are different mappings, for example , then each map that maps and onto different elements in , does what it needs. ${\displaystyle K}$ ${\displaystyle {\mathcal {Set}}}$ ${\displaystyle f,g:A\rightarrow B}$ ${\displaystyle f(a)\not =g(a)}$ ${\displaystyle h:B\rightarrow K}$ ${\displaystyle f(a)}$ ${\displaystyle g(a)}$ ${\displaystyle K}$

• In the category of topological spaces , every non-empty discrete space is a generator and every space that contains at least a two-element subspace with the trivial subspace topology is a cogenerator. ${\displaystyle {\mathcal {T\!op}}}$
• In the category of completely regular rooms or in the category of compact Hausdorff rooms , the unit interval is a cogenerator. ${\displaystyle [0,1]}$
• In the category of modules above a ring , the ring understood as a module is a generator. ${\displaystyle {\mathcal {Mod}}_{R}}$ ${\displaystyle R}$ ${\displaystyle R}$
• The category rings with one element does not have any cogenerators. (For if a cogenerator, it would have to be two different Körpermorphismen a morphism be with . But morphisms on bodies are always the zero function or injective , which is why it body with a thickness greater than the thickness of no such can give.) ${\displaystyle R}$ ${\displaystyle K\rightarrow K}$ ${\displaystyle h:K\rightarrow R}$ ${\displaystyle h\circ f\not =h\circ g}$ ${\displaystyle h:K\rightarrow R}$ ${\displaystyle R}$ ${\displaystyle h:K\rightarrow R}$

properties

Hom functors

A simple rewording used as a definition by some authors is:

An object from is a generator for if and only if the Hom functor is true . ${\displaystyle G}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \mathrm {hom} _{\mathcal {C}}(G,-):{\mathcal {C}}\rightarrow {\mathcal {Set}}}$

Dual applies to this

An object from is a cogenerator for if and only if the Hom functor is true. ${\displaystyle K}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \mathrm {hom} _{\mathcal {C}}(-,K):{\mathcal {C}}\rightarrow {\mathcal {Set}}}$

Products and coproducts

The following properties show how generators and co-generators can be related to any objects in the category:

An object of a category that any coproducts owns is, if and only a generator for , if there is any object of a lot and a epimorphism of -fold of coproduct to exist. ${\displaystyle G}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle A}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle I}$ ${\displaystyle \textstyle \coprod _{i\in I}G\rightarrow A}$ ${\displaystyle I}$ ${\displaystyle G}$ ${\displaystyle A}$

Dual applies to this

An object of a category that any products has is, just then a cogenerator for , if for every object from a set and a monomorphism of into the fold of product out there. ${\displaystyle K}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle A}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle I}$ ${\displaystyle \textstyle A\rightarrow \prod _{i\in I}K}$ ${\displaystyle A}$ ${\displaystyle I}$ ${\displaystyle K}$

Individual evidence

1. ↑ B. Pareigis: Categories and Functors , Springer Verlag 1969, ISBN 978-3-519-02210-7 , chap. 2.10 Generators and Co-Generators
2. ^ H. Schubert: Categories I , Springer-Verlag 1970, ISBN 978-3-540-04865-7 , definition 10.5.1 °
3. ^ H. Herrlich, GE Strecker: Category Theory , Ally and Bacon Inc. 1973, Definition 12.18
4. ↑ H. Herrlich, GE Strecker: Category Theory , Ally and Bacon Inc. 1973, Examples 12.21 (12)
5. ↑ H. Herrlich, GE Strecker: Category Theory , Ally and Bacon Inc. 1973, Examples 12.21 (14)
6. ↑ B. Pareigis: Categories and Functors , Springer Verlag 1969, ISBN 978-3-519-02210-7 , chap. 2.10 Generators and Co-Generators , Lemma 2
7. ^ H. Herrlich, GE Strecker: Category Theory , Ally and Bacon Inc. 1973, sentence 19.6