Regular monomorphism and epimorphism
Regular monomorphisms and epimorphisms are terms from the mathematical subfield of category theory . It is a tightening of the monomorphisms or epimorphisms .
definition
A morphism in a category is called regular monomorphism if it is a difference kernel, that is, if there are morphisms such that there is a difference kernel of and .
Dual definition is:
A morphism in a category is called regular epimorphism if it is a difference coker, that is, if there are morphisms such that there is a difference coker of and .
Note that difference kernels are always monomorphisms and difference kernels are always epimorphisms, so that this is actually a tightening of the terms mono- and epimorphism.
Examples
- In the categories of the quantities of the groups , the - left modules over a ring or in the category of compact Hausdorff spaces all monomorphisms and epimorphisms are automatically regular. So the term doesn't bring anything new here.
- In the category of topological spaces with the continuous mappings , the regular monomorphisms are exactly the homeomorphisms on the picture and the regular epimorphisms are exactly the topological quotient maps . In this category you can easily find monomorphisms or epimorphisms that are not regular.
- In the category of rings with one and the ring homomorphisms , which map the single element back onto the single element, the inclusion mapping is a monomorphism that is not regular.
- Retractions are regular epimorphisms, coretractions are regular monomorphisms.
Remarks
- Regular monomorphisms and regular epimorphisms are extreme .
- Compositions of regular monomorphisms (or epimorphisms) are generally not regular.
Individual evidence
- ↑ Martin Brandenburg: Introduction to Category Theory , Springer-Verlag (2016), ISBN 978-3-662-53520-2 , definition 6.7.22
- ^ Horst Herrlich, George E. Strecker: Category Theory , Allyn and Bacon Inc. 1973, definition 16.13
- ↑ Maria Cristina Pedicchio, Walter Tholen (ed.): Categorical Foundations , Cambridge University Press (2004), Chapter IV, Definition 2.16
- ↑ Horst Herrlich, George E. Strecker: Category Theory , Allyn and Bacon Inc. 1973, examples 16.14
- ↑ Horst Herrlich, George E. Strecker: Category Theory , Allyn and Bacon Inc. 1973, sentence 16.15
- ↑ Horst Herrlich, George E. Strecker: Category Theory , Allyn and Bacon Inc. 1973, sentence 17.11