Subsystems and regular quotients of C-systems

TL;DR

This paper investigates sub-objects and regular quotient-objects of C-systems, focusing on their properties and structure, contributing to the understanding of categorical models in type theory.

Contribution

It provides a detailed analysis of sub-objects and regular quotients in C-systems, expanding the theoretical framework of contextual categories.

Findings

Characterization of all sub-objects in C-systems

Analysis of regular quotient-objects and their properties

Identification of conditions for surjective projection morphisms

Abstract

C-systems were introduced by J. Cartmell under the name "contextual categories". In this note we study sub-objects and quotient-objects of C-systems. In the case of the sub-objects we consider all sub-objects while in the case of the quotient-objects only {\em regular} quotients that in particular have the property that the corresponding projection morphism is surjective both on objects and on morphisms. It is one of several short papers based on the material of the "Notes on Type Systems" by the same author. This version is essentially identical with the version published in Contemporary Mathematics n.658.