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.
Highlights
The purpose of this study is to extend the method of taking the real spectrum or its analogues to the case of infinite gluings of the small spaces considered in real algebraic or analytic geometry or in model theory and to make another step in building general topology for locally small spaces, which can be considered as topological spaces with additional structure
The main result of the paper reads as follows: the category of Kolmogorov locally small spaces and bounded continuous mappings is equivalent to the category of spectral spaces with distinguished decent lumps and with bornologies in the lattices of compact open sets as objects and spectral mappings respecting those decent lumps and satisfying a boundedness condition as morphisms and 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 satisfying a domination condition and respecting those decent lumps as morphisms
We proved new versions of Stone Duality (Theorems 1, 2, 6 and 7) and gave an equivalent description of the category of up-spectral spaces and their spectral mappings (Theorem 5), giving new instances of symmetry on the category theory level
Summary
We have another version of Stone Duality: for Kolmogorov locally small spaces with bounded strongly continuous mappings This category is equivalent to the category of up-spectral spaces with distinguished patch dense subsets as objects and strongly spectral mappings respecting those patch dense subsets as morphisms and is dually equivalent to the category of distributive lattices with zeros and distinguished patch dense sets of prime filters as objects and lattice homomorphisms respecting zeros and those patch dense sets and satisfying a condition of domination as morphisms. These new versions of Stone Duality give more understanding of objects and morphisms of the categories we introduce.
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