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Etale algebras over finite Heyting algebras

Kuznetsov Evgeny

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.

Etale algebras over finite Heyting algebras

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.

Paper Structure

This paper contains 9 sections, 21 theorems, 11 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Stone duality and Esakia duality
  3. Local homeomorphisms
  4. Local homeomorphisms of topological spaces
  5. Ordered and ordered topological analogues
  6. Étale Heyting algebras
  7. Equivalencies and dualities for étale algebras over a finite Heyting algebra
  8. Conclusion and future work
  9. General case

Key Result

Lemma 1.6

In a Priestley space $X$ the subsets ${\uparrow} x$ and ${\downarrow} x$ are closed subsets for each $x\in X$.

Figures (2)

  • Figure 1: Map of main results
  • Figure 2: Pullback of $\alpha, \beta$

Theorems & Definitions (48)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Lemma 1.6
  • Lemma 1.7
  • Definition 1.8
  • Definition 1.9
  • Definition 2.1
  • ...and 38 more