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 . ${\displaystyle e:Z\rightarrow X}$ ${\displaystyle f,g:X\rightarrow Y}$ ${\displaystyle e}$ ${\displaystyle f}$ ${\displaystyle g}$

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 . ${\displaystyle e:Y\rightarrow Z}$ ${\displaystyle f,g:X\rightarrow Y}$ ${\displaystyle e}$ ${\displaystyle f}$ ${\displaystyle g}$

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. ${\displaystyle R}$ ${\displaystyle R}$
• 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. ${\displaystyle \mathbb {Z} \hookrightarrow \mathbb {Q} }$
• 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

1. ↑ Martin Brandenburg: Introduction to Category Theory , Springer-Verlag (2016), ISBN 978-3-662-53520-2 , definition 6.7.22
2. ^ Horst Herrlich, George E. Strecker: Category Theory , Allyn and Bacon Inc. 1973, definition 16.13
3. ↑ Maria Cristina Pedicchio, Walter Tholen (ed.): Categorical Foundations , Cambridge University Press (2004), Chapter IV, Definition 2.16
4. ↑ Horst Herrlich, George E. Strecker: Category Theory , Allyn and Bacon Inc. 1973, examples 16.14
5. ↑ Horst Herrlich, George E. Strecker: Category Theory , Allyn and Bacon Inc. 1973, sentence 16.15
6. ↑ Horst Herrlich, George E. Strecker: Category Theory , Allyn and Bacon Inc. 1973, sentence 17.11