Abstract

We study the algebras for the double power monad on the Sierpinski space in the cartesian closed category of equilogical spaces and produce a connection of the algebras with frames. The results hint at a possible synthetic, constructive approach to frames via algebras, in line with that considered in Abstract Stone Duality by Paul Taylor and others.

Highlights

  • The category Equ of equilogical spaces introduced by Dana Scott in [24] offers a very nice extension of the category Top0 of T0 -spaces and continuous maps, as it is a locally

  • There is a problem in reading the previous sentence: the object ST need not exist as a topological space

  • Bucalo and Rosolini [5] suggested that the algebras for the monad S2 resembled frames with frame homomorphisms and we address exactly that connection in the present paper

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Summary

Introduction

The category Equ of equilogical spaces introduced by Dana Scott in [24] offers a very nice extension of the category Top0 of T0 -spaces and continuous maps, as it is a locally. The immediate solution is to read that sentence in the category of equilogical spaces where ST always exists; it is just that it may be a true equilogical space (ie which is not topological) Such an extension of the language of Cartesian closed categories (and of the λ-calculus) was tested in various guises in many papers, see for instance Taylor [28, 32] and Vickers and Townsend [34]. In particular we show that every S2 –algebra is an internal frame in Equ in Corollary 5.7 and show how their global sections are connected to frames in Corollary 5.8

Basic properties of equilogical spaces
The monad of the double power
The algebraic theory of an object
Global sections of S2–algebras
Final remarks and further directions
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