2008/02/05 by Claudio Pisani, Pisani, Claudio
Mathematics · #18A99 #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.CT #msc:18A99
paper · pdf · doi:10.48550/arxiv.0802.0600
32 pages, corrected typos and minor changes
openalex publication_date 2008/02/05 · arxiv created 2008/02/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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→\Cat, which preserve (or reverse) slices and adjunctible maps and which can be enriched over internal sets too.