# Some aspects of (non) functoriality of natural discrete covers of locales

@article{Ball2019SomeAO, title={Some aspects of (non) functoriality of natural discrete covers of locales}, author={R. Ball and J. Picado and A. Pultr}, journal={Quaestiones Mathematicae}, year={2019}, volume={42}, pages={701 - 715} }

Abstract The frame Sc(L) generated by closed sublocales of a locale L is known to be a natural Boolean (“discrete”) extension of a subfit L; also it is known to be its maximal essential extension. In this paper we first show that it is an essential extension of any L and that the maximal essential extensions of L and Sc(L) are isomorphic. The construction Sc is not functorial; this leads to the question of individual liftings of homomorphisms L → M to homomorphisms Sc(L) → Sc(M). This is… Expand

#### 3 Citations

ON JOINS OF COMPLEMENTED SUBLOCALES

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The system Sc(L) consisting of joins of closed sublocales of a locale L is known to be a frame, and for L subfit it coincides with the Booleanization Sb(L) of the coframe of sublocales of L. In this… Expand

Exact Filters and Joins of Closed Sublocales

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We prove, for a general frame, that the sublocales that can be represented as joins of closed ones are, somewhat surprisingly, in a natural one-to-one correspondence with the filters closed under… Expand

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