On Equalizers in the Category of Locales
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On Equalizers in the Category of Locales Jorge Picado1
· Aleš Pultr2
Received: 29 June 2020 / Accepted: 25 October 2020 © Springer Nature B.V. 2020
Abstract The fact that equalizers in the context of strongly Hausdorff locales (similarly like those in classical spaces) are closed is a special case of a standard categorical fact connecting diagonals with general equalizers. In this paper we analyze this and related phenomena in the category of locales. Here the mechanism of pullbacks connecting equalizers is based on natural preimages that preserve a number of properties (closedness, openness, fittedness, complementedness, etc.). Also, we have a new simple and transparent formula for equalizers in this category providing very easy proofs for some facts (including the general behavior of diagonals). In particular we discuss some aspects of the closed case (strong Hausdorff property), and the open and clopen one. Keywords Frame · Locale · Sublocale · Localic map · Image and preimage · Binary product of locales · Strongly Hausdorff locale · Diagonal map · Equalizer · Open diagonal Mathematics Subject Classification 18F70 · 06D22
Introduction The diagonal morphism is an equalizer of the projections, and pullbacks pull back equalizers, as in the following diagram
The authors gratefully acknowledge financial support from the Centre for Mathematics of the University of Coimbra (UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES) and from the Department of Applied Mathematics (KAM) of Charles University (Prague).
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Jorge Picado [email protected] Aleš Pultr [email protected]
1
Department of Mathematics, University of Coimbra, CMUC, 3001-501 Coimbra, Portugal
2
Department of Applied Mathematics and ITI, MFF, Charles University, Malostranské nám. 24, 11800 Praha 1, Czech Republic
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J. Picado, A. Pultr
P
Equ( f 1 , f 2 )
g
B
δB
/AK KK KK KK fi KK f1 , f2 KK KK KK K% / B×B /B p i
This simple categorical fact covers a number of facts connecting the behaviour of diagonals in categories with the general behaviour of equalizers (a very special example being the closed equalizers in Hausdorff spaces related to the fact that the Hausdorff property of X is characterized by the closed diagonal in X × X , for more see [2]). In this paper we concentrate on some aspects of this phenomenon in the category of locales. In this category we can exploit some expedient concrete facts: subobjects, the so called sublocales, are very transparent entities, easy to work with, pulling back is being done by preimages, again transparent and closely reminiscent to the classical preimages both in the form and capability (preserving closedness and openness and other useful properties, some of them more relevant in the point-free context then classically), and a new very simple formula for equalizer. In Preliminaries we recall the standard notation and facts, and add, for convenience of the reader, a brief description of binary coproducts of frames in the form it will be used. Then, we interpret the ment
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