Skip to main content Accessibility help
×
Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-22T15:57:23.737Z Has data issue: false hasContentIssue false

IV - Regular, Protomodular, and Abelian Categories

Published online by Cambridge University Press:  05 November 2013

Dominique Bourn
Affiliation:
Université du Littoral
Marino Gran
Affiliation:
Université du Littoral
Maria Cristina Pedicchio
Affiliation:
Università degli Studi di Trieste
Walter Tholen
Affiliation:
York University, Toronto
Get access

Summary

The aim of this chapter is mainly to introduce some basic categorical concepts dealing with General Algebra, and to illustrate in this context many important properties of one of the most remarkable categories in this area, namely the category Grp of groups.

These concepts are basic in the sense that they deal with notions as elementary as, for instance, those of epimorphism, monomorphism, kernel, cokernel and equivalence relation.

The first section is devoted to setting basic facts concerning the internal equivalence relations in a given finitely complete category.

In the second section different notions of epimorphisms are considered and classified, and their stability properties are studied. We prove a new result concerning strong epimorphisms, which turns out to be a powerful tool in many situations. We call this result the Barr-Kock Theorem, in reference to an important and weaker version of it given in [1]. The notions of regular and Barr-exact categories are then recalled, and their main basic properties are proved. In this context the previous distinctions of different kinds of epimorphisms are pretty well simplified.

In Algebra, among the several notions of monomorphism (or, equivalently, of subobject), the class of normal subgroups in the category Grp of groups, or the class of ideals in the category Rng of rings naturally deserve great attention. The third section deals with the general notion of normal monomorphism. It introduces the concept of protomodular category, where this notion of normal monomorphism becomes intrinsic, precisely as this happens in Grp or Rng.

Type
Chapter
Information
Categorical Foundations
Special Topics in Order, Topology, Algebra, and Sheaf Theory
, pp. 165 - 212
Publisher: Cambridge University Press
Print publication year: 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×