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Category Archives: Uncategorized
Sheaves and streams II: sheafification, and stratified étale maps
In part I, I explained how one can build the étale space of a presheaf F over a topological space X. I will show how one can retrieve a sheaf from an étale map, leading to a nice adjunction and … Continue reading
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Sheaves and streams I: sheaves of locally monotone maps
Sheaves are a fundamental notion. In this post and later posts, I would like to explain some of the basic theory of the most mundane notion of sheaves: sheaves of sets over a topological space. My real goal is really … Continue reading
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Statures of Noetherian spaces I: maximal order types of wpos
I have had a very gifted masters 2 student from midMay to late July, Bastien Laboureix. He mostly solved the questions I had left open here. I wanted to report on his work, but that is a lot too technical … Continue reading
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Topological functors I: definition, duality, limits and colimits
I have briefly mentioned topological functors in a recent post. It is time for me to explain what they are. This is a truly wonderful concept, which abstracts topological spaces away and concentrates on the key properties of the forgetful … Continue reading
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From prestreams to streams
Last time, I had described two satisfactory models of topological spaces with a local direction of time: Marco Grandis’ dspaces, and Sanjeevi Krishnan’s prestreams. The two kinds form categories that are related by an adjunction S ⊣ D, discovered by … Continue reading
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Prestreams and dspaces
How do you model a topological space with a direction of time? That should seem easy; for example, a topological space with a preordering should be enough. But how do you model the directed circle, where times goes counterclockwise? That … Continue reading
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Firstcountable spaces and their Smyth powerdomain
This month, we will look at certain conditions recently found by He, Li, Xi and Zhao in 2019, and then by Xu and Yang in 2021, in order to ensure that the Smyth powerdomain Q(X) (with the Scott topology) of … Continue reading
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The Sorgenfrey line is not consonant
In Exercise 5.4.12 of the book, I ask the reader to prove that neither the space of rationals, Q, nor the Sorgenfrey line, Rℓ, is consonant. But the proofs I had in mind were much too simpleminded to stand any … Continue reading
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Bitopological spaces and stable compactness
A while back (in March 2019, to be precise), Tomáš Jakl told me that he had a nice, short proof of the fact that the categories of stably compact spaces (and perfect maps) and compact pospaces (and continuous orderpreserving maps) … Continue reading
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Plotkin’s powerdomain and the hedgehog
There are three classical powerdomains in domain theory, named after Hoare, Smyth, and Plotkin. The first two are natural and well studied, and the third one is intricate and intriguing. To start with, there are several possible definitions for a … Continue reading
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