Categories and General Algebraic Structures with Applications

Categories and General Algebraic Structures with Applications

Localic maps constructed from open and closed parts

Document Type : Research Paper

Authors
Ales Pultr 1
Jorge Picado 2
1 Department of Applied Mathematics and ITI, MFF, Charles University, Malostransk'e n'am. 24, 11800 Praha 1, Czech Republic.
2 CMUC, Department of Mathematics, University of Coimbra, Apar\-ta\-do 3008, 3001-501 Coimbra, Portugal.
Abstract
Assembling a localic map $f\colon L\to M$ from localic maps $f_i\colon S_i\to M$, $i\in J$, defined on closed resp. open sublocales $(J$ finite in the closed case$)$ follows the same rules as in the classical case. The corresponding classical facts immediately follow from the behavior of  preimages but for obvious reasons such a proof cannot be imitated in the point-free context. Instead,  we present  simple proofs based on categorical reasoning. There are some related aspects of localic preimages that are of interest, though. They are investigated in the second half of the paper.

Highlights

Dedicated to Bernhard Banaschewski on the occasion of his 90th birthday

Keywords
frame
locale
sublocale
sublocale lattice
open sublocale
closed sublocale
localic map
preimage
Boolean frame
linear frame
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