Pointfree convergence approach and pointfree index theory
Abstract
Given a category C of lattices, we introduce the category CCap, a pointfree version of the category Cap of convergence approach spaces, that contains Cap coreflectively, as well as the category CConv, 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 CConv, the category Ap of approach spaces finds a natural analog CAp in CCap 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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BibTeX
@misc{anguloperkins2026pointfree,
title = {{Pointfree convergence approach and pointfree index theory}},
author = {Emilio Angulo-Perkins and Frédéric Mynard},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05621},
eprint = {2611.05621},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05621},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Pointfree convergence approach and pointfree index theory
AU - Emilio Angulo-Perkins
AU - Frédéric Mynard
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05621
UR - https://arxiv.org/abs/2611.05621
AB - 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.
ER -