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 due to Anders Kock which he calls KZ-monads. My purpose is to talk about Escardó’s argument; but mostly, really, to put forward his notion of KZ-monad, which is a true categorical gem. Read the full post.
In 1969, Ernest Manes proved the following remarkable result: the algebras of the ultrafilter monad on Set are exactly the compact Hausdorff spaces. This is remarkable, because it gives a purely algebraic/category-theoretic of the otherwise purely topological notion of compact Hausdorff spaces. I will explain how this is proved, and I will give a few pointers to some extension results of the same kind. Read the full post.
I have been attending the 9th International Symposium on Domain Theory (ISDT’22), which took place online, July 4-6, 2022, in Singapore. This was a fine conference indeed, and it ran very smoothly. I initially intended to give a summary of what happened there, but in the end I decided to concentrate on just two contributed papers: one by Hualin Miao, Qingguo Li, and Dongsheng Zhao, about those posets that have one-step closure, namely in which one can compute the Scott-closure of a downwards-closed subset A by just taking the collection of suprema of directed families included in A; the other one by Zhenchao Lyu, Xiaolin Xie, and Hui Kou, who showed that the category of c-spaces and the category of locally finitary compact spaces are not Cartesian-closed, by an argument that is both easy and clever. Read the full post.
I have already given an argument for the non-consonance of the Sorgenfrey line Rℓ here. I would now like to explain why the space Q of rational numbers is not consonant either. That is quite a challenge. The most easily accessible proof is due to Costantini and Watson, but it still requires some effort to understand. Fortunately, the topological game of last time will help us make sense of at least one half of the construction. Read the full post.
Showing that Q is not consonant is quite an ordeal. I have finally managed to understand one of the existing proofs of this fact, due to Costantini and Watson. This would be a bit too long to cover entirely in one post, so the bulk of the explanation will be for another time. Instead, I will explain why the compact subsets of Q are all scattered, and what it means, but the important point of this month’s post is that, reading between the lines, the Costantini-Watson argument relies on a property that I will characterize through the use of a topological game G(K), resembling the strong Choquet game, in which we will see that player I has a winning strategy if K is compact and scattered—and that is an if and only if in any T2 space. Read the full post.
I would like to explain a clever counterexample due to Jimmie Lawson in 1970, or rather a slight variant of it, pertaining to the theory of topological semilattices and to a property that crops up naturally, namely having small semilattices. Before I can do this, I will have to spend some time explaining what topological semilattices are, and how small semilattices arise naturally. For motivational purposes, I will consider the problem of characterizing the algebras of the so-called finitary Smyth hyperspace monad, a question that Andrea Schalk has solved, among others, in her 1993 PhD thesis. Read the full post.
I would like to talk about a nifty, recent result due to Yu Chen, Hui Kou, and Zhenchao Lyu. There are two natural topologies on the Hoare hyperspace of a space X, the Scott and the lower Vietoris topology, and one may wonder when they coincide. Outside of the realm of posets in their Scott topology, we will see that they rather rarely coincide. The result that is the core of this month’s post is that they do if X is a poset (with its Scott topology) satisfying what I will call the Chen-Kou-Lyu property; and that this property holds if the poset X is core-compact, or first-countable. Read the full post.
Stone duality relates topological spaces and locales (or frames). But there are really many sorts of Stone dualities. In 1997, Yixiang Chen studied Stone dualities that relate so-called L-domains to so-called distributive D-semilattices. This was refined later in a common paper with Achim Jung. This is a very nice theory, which looks a lot like ordinary Stone duality between topological spaces and frames, but with a few twists. As we will see, the resulting monad, which I would like to call algebraicization, turns every L-domain into an algebraic L-domain. Read the full post.
A ∩-semilattice of sets is a family of sets that is closed under finite intersections, and it is irredundant if and only if all its non-empty elements are irreducible. That sounds like a ridiculously overconstrained notion, but I will give two applications of the notion. One, which I will actually present last, is a baby version of the so-called Groemer theorem. This baby Groemer theorem is non-trivial, but has an amazingly simple proof, due to Klaus Keimel; we have used it to prove non-Hausdorff generalizations of a line of theorems due to Choquet, Kendall and Matheron. The other application is due to Zhenchao Lyu and Xiaodong Jia. The Smyth powerdomain Q(X) of a space X is locally compact if and only if X is, and they were interested in knowing whether the same would happen with “core-compact” instead of “locally compact”. The answer is no, and this rests on a clever use of irredundancy, and the existence of a core-compact, non-locally compact space. Read the full post.
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 its associated monad, sheafification. This is all well-known, but then I would like to apply all that to the presheaf of locally monotone functions of a prestream, which we had already started to examine last time. We will obtain a funny structure that I will call stratified étale maps. Read the full post.