Aspects of slice stability in locale theory (Q2897183)

An axiom, placed on a category \(\mathcal C\), is said to be \textit{slice-stable} if, whenever it is satisfied by \(\mathcal C\), it is also satisfied by the slice category \({\mathcal C}/X\) for every object \(X\) of \(\mathcal C\). In this long and interesting paper the author shows that the axiomatic accounts of the category of locales developed by him in a series of three papers [Math. Proc. Camb. Philos. Soc. 139, No. 3, 441--455 (2005; Zbl 1104.06008); Commentat. Math. Univ. Carol. 48, No. 3, 541--553 (2007; Zbl 1199.06039); J. Pure Appl. Algebra 214, No. 6, 729--739 (2010; Zbl 1274.18010)] are all slice-stable (modulo a few trivial modifications that do not affect the results). This localic slice stability is then used to give new proofs ofNEWLINENEWLINE\noindent (1) the fundamental theorem of topos theory, which asserts that the axioms of an elementary topos are slice-stable [\textit{P. T. Johnstone}, Sketches of an elephant. A topos theory compendium. I. Oxford: Clarendon Press (2002; Zbl 1071.18001), Section A2.3], and ofNEWLINENEWLINE\noindent (2) \textit{A. Joyal} and \textit{M. Tierney}'s result on the slice stability of locales [``An extension of the Galois theory of Grothendieck'', Mem. Am. Math. Soc. 309, 71 p. (1984; Zbl 0541.18002)].NEWLINENEWLINEAfter an introductory section, Section 2 contains some categorical background on order-enriched categories and categorical change of base, crucial for the proofs in the paper. Section 3 recalls the axioms to be discussed and is devoted to show that they are all slice-stable. In particular, it then follows that the Hofmann-Mislove theorem is slice-stable. As an application of this, the author proves in Section 4 the known result that discrete objects are exponentiable. Then the ideal completion of a preorder is introduced axiomatically. It is shown that, when acting on semilattices, the ideal completion construction is functorial. Finally, in Section 5, the new proofs of the aforementioned fundamental theorem of topos theory and Joyal-Tierney's result are given. The paper ends with a section containing some concluding comments and a table summarizing the axioms (with some comments on them), and an appendix providing one further example of a slice-stable axiom (concerning the product as join-semilattice tensor); as a consequence, all of the results of [Zbl 1199.06039] are slice-stable.