# Pointfree convergence approach and pointfree index theory

“Pointfree convergence approach and pointfree index theory” (2026), by Emilio Angulo-Perkins et al. Computer Science research paper. Original abstract below, unchanged. A free PDF is available.

- Authors: Emilio Angulo-Perkins, Frédéric Mynard
- Published: November 3, 2026
- Field: Computer Science
- DOI: https://doi.org/10.48550/arxiv.2611.05621
- Free PDF: https://arxiv.org/pdf/2611.05621

## Abstract

Given a category C of lattices, we introduce the category C^Cap, a pointfree version of the category Cap of convergence approach spaces, that contains Cap coreflectively, as well as the category C^Conv, of pointfree convergence spaces introduced by Goubault-Larrecq and Mynard, both reflectively and coreflectively. We investigate pointfree counterparts of pseudo-approach, pre-approach, and approach spaces along two natural axes---depth and adherence-determinedness---which coincide in the spatial case but diverge in general lattices, and we determine when the point--powerset adjunction restricts to these subcategories. Unlike in C^Conv, the category Ap of approach spaces finds a natural analog C^Ap in C^Cap and "index theory" translating traditional topological or metric results into generalizing inequalities can be conducted. We illustrate this with a result involving measure of continuity and measure of compactness.

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