SAMS Subject Classification Number: 13, 4

Recall that the Hausdorff property of a topological space $X$ is characterized by the closedness of the diagonal in $X\times X$. This is a general phenomenon, the so-called $\mathcal{P}$-separation [2]:

Given a property relevant in the category in question (typically of a topological nature), an object $X$ is $\mathcal{P}$-separated if the diagonal in $X\times X$ has the property $\mathcal{P}$. Besides the Hausdorff property in classical spaces, relevant examples for $\mathcal{P}$ are e.g. the strong Hausdorff axiom or the Boolean property in the category of locales [4,6].

In the context of locales there are the important properties of fittedness and fitness [5]. A sublocale of a locale is fitted if it is an intersection of open ones and a locale is fit if each of its sublocales is fitted. Since the intersection $S^{\circ }=\bigcap \{T\mid S\subseteq T,T\text{ open}\}$ is an operation of closure type [3], it is natural to ask about fitted diagonals; we will speak of the $\mathcal{F}$-separated locales [1]. This property will be the main topic of this talk.

Taking into account the fact that the subcategory of fit locales is closed under products and subobjects, we have an immediate observation that fitness implies $\mathcal{F}$-separatedness. We will see that $\mathcal{F}$-separatedness is in fact strictly weaker than fitness and we will explore a surprising parallel with the strong Hausdorff axiom, including a Dowker-Strauss type theorem and a characterization in terms of certain relaxed morphisms.

[1] I. Arrieta, J. Picado and A. Pultr, A new diagonal separation and its relations with the Hausdorff property, Appl. Categ. Structures 30 (2022) 247–263.

[2] M.M. Clementino, E. Giuli and W. Tholen, A functional approach to general topology, in: Categorical Foundations, Encyclopedia Math. Appl., vol. 97, Cambridge Univ. Press, Cambridge, 2004, pp. 103–163.

[3] M.M. Clementino, J. Picado and A. Pultr, The other closure and complete sublocales, Appl. Categ. Structures 26 (2018) 891–908.

[4] A. Joyal and M. Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 51 (1984), no. 309.

[5] J. Picado and A. Pultr, Separation in point-free topology, Birkhäuser-Springer, Cham, 2021.

[6] J. Picado and A. Pultr, On equalizers in the category of locales, Appl. Categ. Structures 29 (2021) 267–283.