Some aspects of basic category theory are developed in a finitely complete category $\cal C$, 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 $\cal C$, and several classical properties concerning them can be effectively proved. For any object $X$ of $\cal C$, by restricting $\cal C/X$ 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 $\cal C$: 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 $\cal C \to Cat$, which preserve (or reverse) slices and adjunctible maps and which can be enriched over internal sets too.
Keywords: factorization systems, final maps and discrete fibrations, initial maps and discrete opfibrations, reflections, internal sets and components, slices and coslices, colimiting cones, adjunctible maps, dense maps, cylinders and homotopy, arrow intervals, enrichment, complements
2000 MSC: 18A99
Theory and Applications of Categories,
Vol. 20, 2008,
No. 6, pp 85-115.