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Electronic supplements to the book

Dear all, the site has moved… there (follow the link). Sorry for the inconvenience.

If you had subscribed before Monday, November 27th, 2023, 15h UTC+1, then you are still subscribed on the new site, and you have nothing else to do.

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We pursue last month’s post, written with Zhenchao Lyu, and we describe Gierz’s example of a compact semilattice — namely, a compact Hausdorff topological semilattice — which does not have small semilattices. It is a bit simpler than Jimmie Lawson’s counterexample, which was already described on this blog. Read the full post.

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I have already talked about compact semilattices before, but there is a lot more to say, especially on the subject of having small semilattices or not. Zhenchao Lyu is joining me this month, and we will pursue this next month. We start with Jimmie Lawson’s characterization of those compact semilattices that have small semilattices as being those that have enough interval homomorphisms. We will then exploit that and show that, once we know just one compact semilattice with small semilattices (for example, one given by J. Lawson himself), then we can produce a whole lot of them, using products or the Hoare hyperspace construction. Read the full post.

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Tagged compactness, continuous dcpo, Hoare hyperspace, powerdomain, semilattice, Urysohn
Comments Off on Compact semilattices without small semilattices I: interval homomorphisms, products, and the Hoare hyperspace

There is a well-known formula in domain theory, which, given a monotonic map *f* from a basis *B* of a continuous poset *X* to a dcpo *Y*, produces the largest continuous map *f’* defined on the whole of *X* and below *f* on *B*. This is *Scott’s formula*: *f’*(*x*) ≝ sup_{b ∈ B, b ≪ x} *f*(*b*). Generalizing this to the setting where *X* is a c-space, I will explain when exactly *f’* coincides with *f* on *B*, hence when *f’* is a continuous extension of *f*. Also, using a motivation from the theory of continuous valuations, I will describe conditions under which any algebraic laws satisfied by *f* transfer to *f’*. Read the full post.

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Bounded-complete domains, or bc-domains, are an amazingly rich kind of continuous domains. They form a Cartesian-closed category, and they are the densely injective topological spaces, among other properties. One characterization of bc-domains which I have not included in the book is that they are related in a very precise sense to Lawson semilattices, namely to compact semilattices with small semilattices. This is the *fundamental theorem of compact semilattices*, which I will (re)prove by relying a lot on the theory of compact pospaces and stably compact spaces: see the full post.

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Tagged compact pospace, semilattice
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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 1979. Read the full post.

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Tagged continuous lattice, exponentiability, locale
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Joint continuity is a stronger property than separate continuity. In what cases are those properties equivalent? The question was solved, partially, by Yuri Ershov in 1997, and completely by Bernhard Banaschewski in 1977 (apparently with a gap in the proof) and by Jimmie Lawson in 1985. The answer has to do with locally finitary compact spaces: read the full post.

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Tagged c-space, locally finitary compact, separate continuity
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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 true, as was proved by Martin Hyland in 1979 (published in 1981). I will concentrate on one half of the this result for this time, and I will explain why every exponentiable locale must be a continuous dcpo. The proof is very close to the similar result in **Top**, but, as usual, locales are so much more abstract that similar arguments tend to be harder to understand in **Loc**; I will do my best. Read the full post.

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Tagged cartesian closeness, category theory, continuous lattice, exponentiability, locale
Comments Off on Exponentiable locales I: every exponentiable locale is continuous

Recently, Achim Jung sent me a message from Jimmie Lawson, and suggested that I might be interested in posting the information on this blog. The red book [1] is a precious source of information on domain theory, and if you are interested in knowing how the material there came to be discovered, Continue reading

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Some time ago, I gave an introduction to topological functors. They form a pretty brilliant categorical generalization of topological spaces. The point of today’s post is to give one particular example of the fact that you can somehow generalize some results on topological spaces to topological functors. I will concentrate on showing that a (pretty amazing) construction of certain Cartesian-closed full subcategories of **Top**, due to Martín Escardó, Jimmie Lawson, and Alex Simpson, generalizes pretty smoothly to a pretty large class of topological functors—the so-called well-fibered topological constructs. More precisely, I will concentrate on the first part of this construction, which builds a Cartesian-closed category **Map**_{C} out of a so-called strongly productive class C of objects of a category **C** that forms the domain of a topological construct with discrete terminal objects. Read the full post.

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Tagged cartesian closeness, category theory, exponentiability, topological functor
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