Introducing pixelation with applications

TL;DR

This paper introduces pixelation as a new form of localization in category theory, demonstrating its applications in algebraic topology, sheaf theory, and higher Auslander algebras, offering a novel approximation method.

Contribution

It defines pixelation as a localization technique for categories, connecting it to representation theory, topology, and algebra, and explores its applications in various mathematical structures.

Findings

Pixelation provides a useful approximation of categories.

Application to Zariski topology offers new insights.

Constructs a categorical generalization of higher Auslander algebras.

Abstract

Motivated by the desire for a new kind of approximation, we define a type of localization called pixelation. We present how pixelation manifests in representation theory and in the study of sites and sheaves. A path category is constructed from a set, a collection of "paths" into the set, and an equivalence relation on the paths. A screen is a partition of the set that respects the paths and equivalence relation. For a commutative ring, we also enrich the path category over its modules (=linearize the category with respect to the ring) and quotient by an ideal generated by paths (possibly 0). The pixelation is the localization of a path category, or the enriched quotient, with respect to a screen. The localization has useful properties and serves as an approximation of the original category. As applications, we use pixelations to provide a new point of view of the Zariski topology of localized ring spectra, provide a parallel story to a ringed space and sheaves of modules, and construct a categorical generalization of higher Auslander algebras of type $A$.