Balanced category theory

TL;DR

This paper develops a balanced category theory framework within finitely complete categories with dual factorization systems, enabling natural definitions of classical categorical concepts and their internal enrichments.

Contribution

It introduces a new axiomatization of category theory concepts using dual factorization systems, extending classical ideas to internal enriched categories.

Findings

Classical categorical concepts are definable within the new framework.

Dual underlying categories can be enriched over internal sets.

The construction yields functors to Cat preserving slice and adjunctible structures.

Abstract

Some aspects of basic category theory are developed in a finitely complete category $\C$, endowed with two factorization systems which determine the same discrete objects and are linked by a simple reciprocal stability law. Resting on this axiomatization of final and initial functors and discrete (op)fibrations, concepts such as components, slices and coslices, colimits and limits, left and right adjunctible maps, dense maps and arrow intervals, can be naturally defined in $\C$, and several classical properties concerning them can be effectively proved. For any object $X$ of $\C$, by restricting $\C/X$ to the slices or to the coslices of $X$, two dual "underlying categories" are obtained. These can be enriched over internal sets (discrete objects) of $\C$: internal hom-sets are given by the components of the pullback of the corresponding slice and coslice of $X$. The construction extends to give functors $\C\to\Cat$, which preserve (or reverse) slices and adjunctible maps and which can be enriched over internal sets too.