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On Equalizers in the Category of Locales

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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.

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Notes

  1. A space is sober if every completely prime filter \({\mathcal {F}}\) in \(\Omega (X)\) (that is, an \({\mathcal {F}}\) such that \(\mathop {\textstyle \bigcup }_{i\in J}U_i\in {\mathcal {F}}\) only if \(U_j\in {\mathcal {F}}\) for some \(j\in J\)) is \(\{U \ | \ x\in U\} \) for some \(x\in X\) – in other words, if every system of open sets that looks like a neighborhood system is really a neighborhood system of a point. For instance every Hausdorff space is sober.

  2. If L is regular then a homomorphism \(h:M\rightarrow L\) is one-to-one whenever \(h(a)=1\) implies that \(a=1\) – see e.g. [12, V.5.6].

  3. This axiom was introduced in [9]; there Isbell speaks of unordered frames (see also [10, III.1.5]).

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Acknowledgements

We would like to thank the Editor, Maria Manuel Clementino, and an anonymous referee for careful reading and helpful comments that have improved much the presentation of this article.

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Authors and Affiliations

  1. Department of Mathematics, University of Coimbra, CMUC, 3001-501, Coimbra, Portugal

    Jorge Picado

  2. Department of Applied Mathematics and ITI, MFF, Charles University, Malostranské nám. 24, 11800, Praha 1, Czech Republic

    Aleš Pultr

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  1. Jorge Picado
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  2. Aleš Pultr
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Correspondence to Jorge Picado.

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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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Picado, J., Pultr, A. On Equalizers in the Category of Locales. Appl Categor Struct 29, 267–283 (2021). https://doi.org/10.1007/s10485-020-09616-8

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