Stone Duality for Kolmogorov Locally Small Spaces

Abstract

In this paper, we prove new versions of Stone Duality. The main version is the following: the category of Kolmogorov locally small spaces and bounded continuous mappings is equivalent to the category of spectral spaces with decent lumps and with bornologies in the lattices of (quasi-) compact open sets as objects and spectral mappings respecting those decent lumps and satisfying a boundedness condition as morphisms. Furthermore, it is dually equivalent to the category of bounded distributive lattices with bornologies and with decent lumps of prime filters as objects and homomorphisms of bounded lattices respecting those decent lumps and satisfying a domination condition as morphisms. This helps to understand Kolmogorov locally small spaces and morphisms between them. We comment also on spectralifications of topological spaces.

Esakia Duality for Heyting Small Spaces

“…Recall that each T 0 small space X admits a spectralification (in the sense of the above definition) that is given by applying the functor S Ā from the proof of Theorem 1 (see also Theorem 2) of [6] to X and embedding X into the resulting spectral space (the existence of such an embedding is guaranteed by the functor R of that proof). This spectralification, treating X as a subspace of Y and dropping e in the notation, will be called the standard spectralification of X and denoted by X sp .…”

“…The first tool to analyse small spaces is the theory of spectral spaces, which is already developed enough (see the monograph [13]). The present paper is a continuation of the paper [6] about some versions of Stone Duality ( [14]) or Priestley Duality ( [15]) for locally small spaces. This time, we concentrate on giving a new version of a related duality due to Leo Esakia ( [16]) for small spaces.…”

“…That is why we modify the requirements for a topology. This leads to the notions of a unitary smopology and a small space (already used in [3], [4], [5], [6]) as well as the notions of an arbitrary smopology and a locally small space (used, for example, in [7], [5], [6]). In this way, we construct a kind of algebra-friendly topology.…”

“…Remark 2. By Spec we denote the category of spectral topological spaces (the name introduced by Hochster [17]) and spectral mappings (compare [13], [6]). By the Stone Duality ([13, Chapter 3]), this category is dually equivalent to the category of bounded distributive lattices and their homomorphisms (denoted in [6] by Lat).…”

“…By Spec we denote the category of spectral topological spaces (the name introduced by Hochster [17]) and spectral mappings (compare [13], [6]). By the Stone Duality ([13, Chapter 3]), this category is dually equivalent to the category of bounded distributive lattices and their homomorphisms (denoted in [6] by Lat). Recall that the category of Priestley spaces and Priestley mappings ([13, 1.5.15], denoted here by Pri) is concretely isomorphic to Spec, so these two categories may be identified.…”