Tag Archives: continuous lattice

Exponentiable locales II: the exponentiable locales are the continuous frames

Posted on July 20, 2023 by jgl

Two months ago, we have seen that every exponentiable locale had to be continuous, as a frame. We will see that the converse holds: the continuous frames are exactly the exponentiable locales. The result is due to Martin Hyland in … Continue reading →

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Exponentiable locales I: every exponentiable locale is continuous

Posted on May 21, 2023 by jgl

The exponentiable objects of Top are exactly the core-compact spaces. Through Stone duality, the core-compact spaces are related to the continuous frames. So here is a wild guess: would the exponentiable locales be exactly the continuous frames? That is indeed … Continue reading →

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Algebras of filter-related monads: II. KZ-monads

Posted on September 19, 2022 by jgl

Alan Day and Oswald Wyler once proved that the algebras of the filter monad on the category Top0 of T0 topological spaces are exactly the continuous (complete) lattices. Martín Escardó later gave a very interesting proof of this fact, using a category-theoretic construction … Continue reading →

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Chains and nested spaces

Posted on August 23, 2020 by jgl

A chain is a totally ordered poset, and a nested space is a topological space whose lattice of open sets is a chain. That may seem like a curious notion, although you might say that the Scott topology on the … Continue reading →

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