Etale algebras over finite Heyting algebras

TL;DR

This paper explores etale algebras over finite Heyting algebras, establishing dualities and axiomatizations that simplify computations and deepen understanding of their categorical structure.

Contribution

It introduces etale Heyting H-algebras, establishes duality for finite cases, and provides axioms for their variety, enhancing the theoretical framework.

Findings

Category-theoretic duality for finite etale Heyting H-algebras

Axiomatization of the variety of etale Heyting H-algebras

Simplified computation of finite colimits in the category

Abstract

In this paper, we investigate the concept of local homeomorphism in Esakia spaces. We introduce the notion of etale Heyting H-algebra and establish category-theoretic duality for etale Heyting H-algebra in the case of finite Heyting algebra H. Furthermore, we give an identity that axiomatizes the variety of etale Heyting H-algebras when H is finite. We also show that the category of Stone space-valued (co)presheaves over a finite Esakia space X is equivalent to the slice category of local homeomorphisms over X. The fact is used to show that, in comparison with the case of general Heyting H-algebras, it is easier to compute finite colimits in the category of etale Heyting H-algebras.