A categorical approach to the maximum theorem

TL;DR

This paper generalizes Berge's maximum theorem using monoidal topology and double categories, extending its applicability from topological spaces to various generalized spaces including probabilistic and approach spaces.

Contribution

It introduces a categorical framework that broadens the scope of the maximum theorem to pseudotopological, pretopological, closure, approach, and probabilistic approach spaces.

Findings

Generalized maximum theorem to pseudotopological and pretopological spaces.

Proved a broad version of the extreme value theorem.

Established a categorical foundation for optimization in generalized spaces.

Abstract

Berge's maximum theorem gives conditions ensuring the continuity of an optimised function as a parameter changes. In this paper we state and prove the maximum theorem in terms of the theory of monoidal topology and the theory of double categories. This approach allows us to generalise (the main assertion of) the maximum theorem, which is classically stated for topological spaces, to pseudotopological spaces and pretopological spaces, as well as to closure spaces, approach spaces and probabilistic approach spaces, amongst others. As a part of this we prove a generalisation of the extreme value theorem.