# 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