Balanced Category Theory II

TL;DR

This paper extends the study of balanced factorization categories in category theory, linking internal (co)limits, Cauchy completeness, and topology through a unified abstract framework.

Contribution

It introduces a weak balanced factorization category that synthesizes topological and categorical concepts, revealing deep connections between them.

Findings

Established a link between discrete fibrations and local homeomorphisms.

Demonstrated the correspondence between (co)limits and topological neighborhoods.

Showed how open-closed complementarity aids in categorical internalization.

Abstract

In the first part, we further advance the study of category theory in a strong balanced factorization category C [Pisani, 2008], a finitely complete category endowed with two reciprocally stable factorization systems such that X \to 1 is in M iff it is in M'. In particular some aspects related to "internal" (co)limits and to Cauchy completeness are considered. In the second part, we maintain that also some aspects of topology can be effectively synthesized in a (weak) balanced factorization category T, whose objects should be considered as possibly "infinitesimal" and suitably "regular" topological spaces. While in C the classes M and M' play the role of discrete fibrations and opfibrations, in T they play the role of local homeomorphisms and perfect maps, so that X\to 1 is in M (resp. M') iff it is a discrete (resp. compact) space. One so gets a direct abstract link between the subjects, with mutual benefits. For example, the slice projection X/x \to X and the coslice projection x\X \to X, obtained as the second factors of x:1 \to X according to (E,M) and (E',M') in C, correspond in T to the "infinitesimal" neighborhood of x \in X and to the closure of x. Furthermore, the open-closed complementation (generalized to reciprocal stability) becomes the key tool to internally treat, in a coherent way, some categorical concepts (such as (co)limits of presheaves) which are classically related by duality.