Pointfree convergence approach and pointfree index theory

Abstract

Given a category $\mathsf{C}$ of lattices, we introduce the category $\mathsf{C^{Cap}}$, a pointfree version of the category $\Cap$ of convergence approach spaces, that contains $\Cap$ coreflectively, as well as the category $\mathsf{C^{Conv}}$, 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 $\mathsf{C^{Conv}}$, the category $\mathsf{Ap}$ of approach spaces finds a natural analog $\mathsf{C^{Ap}}$ in $\mathsf{C^{Cap}}$ 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.