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 to explore what sheaf technology can bring us in the study of streams and prestreams. To start with, I will introduce the classical notion of the étale space of a presheaf, and illustrate that on sheaves of locally monotone (resp., and continuous) maps on streams, and particularly in the case of the directed circle. Read the full post.
I have had a very gifted masters 2 student from mid-May 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 for a blog. Instead, I will start a sequence of posts on the notion of stature of a Noetherian space. Since the notion is defined by analogy with a similar notion in the special case of well-partial-orderings, and coincides with something called maximal order types, I will spend this month’s post explaining the latter notion. Read the full post.
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 functor from Top to Set. In other words, that forgetful functor is topological, but there are many others, including some involving streams, prestreams, and d-spaces. We will see some of the classical properties of topological functors, notably that topological functors are self-dual, and that they preserve and create both limits and colimits. Read the full post.
Last time, I had described two satisfactory models of topological spaces with a local direction of time: Marco Grandis’ d-spaces, and Sanjeevi Krishnan’s prestreams. The two kinds form categories that are related by an adjunction S ⊣ D, discovered by Emmanuel Haucourt. S. Krishnan’s purpose was really to talk about streams, not prestreams, and I will show how they emerge from the study of that adjunction. In other words, we will answer the following question partly: what makes the prestreams of the form S(X, dX) so special? Read the full post.
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 should seem harder. Fortunately, there are at least two satisfactory models: Marco Grandis’ d-spaces, and Sanjeevi Krishnan’s (pre)streams, which I will describe. Both appeared in 2009. Although they look rather different, they are connected by an idempotent adjunction, which restricts to an equivalence between so-called complete d-spaces and Haucourt streams, as discovered by Emmanuel Haucourt in 2012. Read the full post.
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 a space X is a dcpo model of X, and in order to ensure that the Scott and upper Vietoris topologies agree on Q(X)… without the usual condition of local compactness, but under some first-countability conditions. I got interested in this while Xiaodong Jia and I explored the Smyth powerdomain of the Sorgenfrey line. Read the full post.
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 simple-minded to stand any chance of succeeding, hence I classified this as important blooper #5 in the list of errata. Good news: Showing that Rℓ is not consonant is not that hard, finally. I will explain the argument in the full post. This will also be an excuse to explain some additional topological properties of Rℓ, an introduction to hereditary Lindelöfness (we will see that Rℓ is hereditarily Lindelöf, although it is not second-countable), and a few additional things in the appendices.
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 order-preserving maps) are equivalent. He uses an approach through bitopological spaces, and this will give me an opportunity to talk about them. Just as with quasi-uniform spaces, I cannot believe it took me so much time before I mentioned bitopological spaces! Read the full post.
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 Plotkin powerdomain, and while all of them are isomorphic in the nice cases of coherent continuous dcpo, or countably-based continuous dcpos, one may wonder whether they would coincide on all continuous dcpos. That is not the case, and I would like to present a funny counterexample, given in an exercise in Abramsky and Jung’s famous Domain Theory chapter. Let me also call it the hedgehog, because it has spines. Read the full post.
“Quasi-continuous domains and the Smyth powerdomain” is the title of a very nice 2013 paper by Reinhold Heckmann and Klaus Keimel. I will not talk about quasi-continuous domains in this post. Rather, I will mention three pearls that this paper contains: one on so-called supercompact sets, a second one now called the topological Rudin Lemma, and finally a pretty surprising characterization of sober spaces that looks a lot like the definition of well-filteredness. Read the full post.