Criteria for homotopic maps to be so along monotone homotopies

TL;DR

This paper develops criteria for when homotopic maps in pospaces can be connected via monotone homotopies, revealing new insights into the structure of time-aware topological spaces and their contractibility.

Contribution

It introduces criteria for homotopies along monotone maps in pospaces and demonstrates future contractibility of hypercontinuous lattices with Lawson topology.

Findings

Hypercontinuous lattices with Lawson topology are future contractible.

Classical homotopy tools can be adapted to structured spacetime models.

Criteria for homotopic maps to be connected via monotone homotopies are established.

Abstract

The state spaces of machines admit the structure of time. A homotopy theory respecting this additional structure can detect machine behavior unseen by classical homotopy theory. In an attempt to bootstrap classical tools into the world of abstract spacetime, we identify criteria for classically homotopic, monotone maps of pospaces to future homotope, or homotope along homotopies monotone in both coordinates, to a common map. We show that consequently, a hypercontinuous lattice equipped with its Lawson topology is future contractible, or contractible along a future homotopy, if its underlying space has connected CW type.