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W1 - Priestley duality and its applications to pointfree topology
Sebastian Melzer, New Mexico State University, United States
This mini course explores Priestley duality for frames. We will begin by recalling the basics of Priestley duality, which provides a topological representation of distributive lattices. Then we will show how this duality naturally extends to frames, offering a powerful toolkit to study frames from a different perspective.
We will investigate key concepts in frame theory—such as spatiality, sublocales, the well-inside and way-below relations—in the language of Priestley spaces. Additionally, we will examine fundamental results, such as the Hofmann-Mislove theorem, and explore important dualities in pointfree topology—including the Hofmann-Lawson and Isbell dualities—all through the lenses of Priestley.
W2 - MCKINSEY-TARSKI (MT)-Algebras
Ranjitha Raviprakash, New Mexico State University, United States
This mini course will offer an alternative view on pointfree topology through the lens of McKinsey-Tarski (MT) algebras. First, we will introduce the category MT of MT-algebras, which are complete boolean algebras equipped with an interior operator, yielding a faithful generalization of the category Top of topological spaces. Moreover, we establish connections between MT and the category Frm of frames, emphasizing the potential of MT as a unifying framework of both Top and MT.
We present a detailed study of separation in the MT-algebras context, extending separation notions from topological spaces and frames. Additionally, we will discuss local compactness within the framework of MT-algebras, which lets us develop an analogue of the Hofmann-Mislove theorem for sober MT-algebras. Using this, we establish versions of Hofmann-Lawson, Isbell, and Stone dualities for MT-algebras. Generalizing the latter two yields new dualities for locally compact Hausdorff and locally Stone spaces.
W3 - Tensor Triangular Geometry in Context
Juan Omar Gómez Rodríguez, Bielefeld University, Germany
Tensor triangular geometry offers a unified framework for addressing ertain classification problems across various fields of mathematics, including algebraic geometry, algebraic topology, and representation theory. This theory bears a strong algebro-geometric flavor: for any essentially small tensor triangulated category, one can associate a spectral space called the Balmer spectrum, constructed analogously to the Zariski spectrum of a commutative ring. The Balmer spectrum parametrizes objects in the category via a specific equivalence relation, making it a powerful invariant that has spurred many recent developments.
The aim of this mini-course is to provide an accessible introduction to tensor triangular geometry. We will review key concepts related to essentially small tensor triangulated categories, then proceed to construct the Balmer spectrum for such a category and explore its properties. Throughout the course, examples from representation theory and commutative algebra will be presented to illustrate these ideas. If time permits, we will also discuss some open questions that could be of interest to point-free topologists.
W4 - The frame of reals, real-valued functions and other maps
Ana Belén Avilez, Chapman University, United States
In this mini course we will focus on the study of the frame of reals, some special embeddings and important results regarding real-valued functions in point-free topology.
First, we will recall the construction of the reals given by Banaschewski in [3] and the notion of continuous real-valued function on a frame L. We will then explore general real-valued functions on a frame (see also [6]) and the powerful tool of scales to build real-valued functions. With this general context, we can then present some types of embedded sublocales (see [2]), the notion of complete separation, and some important insertion, extension and separation results (some of which appear in [4, 5]). If time allows, we would also like to recall the notion of uniform frames and do a similar study for uniform continuous real-valued functions [1].
References
[1] I. Arrieta and A. B. Avilez, A general insertion theorem for uniform locales, J. Pure Appl. Algebra 227 (7) (2923) art. no. 107320.
[2] A. B. Avilez and J. Picado, Continuous extensions of real functions on arbitrary sublocales and C-, C-and z-embeddings, J. Pure Appl. Algebra 225 (2021) art. no. 106702.
[3] B. Banaschewski, The real numbers in pointfree topology, Textos de Matemática, vol. 12, Universidade de Coimbra, 1997.
[4] J. Gutiérrez García and T. Kubiak, General insertion and extension theorems for localic real functions, J. Pure Appl. Algebra 215 (2011) 1198–1204.
[5] J. Gutiérrez García, T. Kubiak and J. Picado, Localic real functions: a general setting, J. Pure Appl. Algebra 213 (2009) 1064–1074.
[6] J. Guti ́errez García, J. Picado and A. Pultr, Notes on point-free real functions and sublocales, in: Categorical Methods in Algebra and Topology, Textos de Matem ́atica, DMUC, vol. 46, pp. 167-200, 2014.
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