8
$\begingroup$

In [1, example C.1.2.8], a locale $Y$ (dense in another locale $X$) without any point is given. I fail to understand the point of such point-less locale - Why can't we identify those as the trivial locales, and what's so great about considering locales that have no points?

Anyway, here's the construction of $X$ and $Y$ (taken from [1]). Let $A$ be an uncountable nonempty set (e.g. $\mathbb{R}$) (equipped with the discrete topology), and let $X$ be the set of all functions $\mathbb{N} \to A$, equipped with the Tychonoff topology. For each $a \in A$, let $X_a$ be the subspace $\{f \in X \,|\, a \in im(f)\}$, and let $$ Y = \bigcap_{a\in A} X_{a}.$$ Now the point set $Y_p$ of $Y$ is empty because there is no onto map from $\mathbb{N}$ to $\mathbb{R}$.

In [2, section 5], Johnstone demonstrates why considering such locales could be useful. The main argument is that topoi are nice things to consider. However, at the point of writing, the (external) applications of topos theory seem lacking. Hopefully the situation has changed in mathematics in recent years. Thus the second question: How does the consideration of pointless locales help topos theory, and how does that in turn applies (externally) to mathematics?

Reference

  • [1] Sketches of an Elephant: A Topos Theory Compendium [Peter T. Johnstone]

  • [2] The point of pointless topology-[Peter T. Johnstone]

$\endgroup$
5
  • 3
    $\begingroup$ If I remember correctly [2] already contains a lot of examples answering your question. For example I assume you have read about the closed subgroup theorem, or the smallest dense sublocales and how an intersection of dense sublocales is dense, or how pointfree locales are tied to set theoretic forcing ? or maybe I'm misunderstanding your question $\endgroup$ Jan 20 at 20:01
  • $\begingroup$ Just to clarify my previous comment : not considering locales without points is perfectly fine, there is no deep reason why you "can't" do it. It justs mean you are no longer working with locales but with (sober) topological spaces. As far as I'm concerned, there is no significant distinction between your question and "what's the point of locales over topological spaces" $\endgroup$ Jan 20 at 20:09
  • 4
    $\begingroup$ The first sentence of the question already contains an answer. If you identify the locale $Y$ there with the trivial locale (as you suggest), then you have a trivial locale dense in a nontrivial space $X$. That conflicts rather badly with the usual meaning of "trivial". $\endgroup$ Jan 20 at 20:14
  • $\begingroup$ Parts of your questions are answered in Sections 1.1f. of this note of mine. Briefly: Allowing locales without points streamlines the theory, helps massively in constructive mathematics (including relative mathematics over a base space) and is crucial for certain representation theorems in topos theory (for instance for showing that for any object $X$ of any topos $E$, there is a surjection $f:F\to E$ such that $f^*(X)$ is countable, namely the classifying $E$-locale of surjections $\mathbb{N}\to X$. This locale doesn't have any points if $X$ is uncountable) $\endgroup$ Jan 21 at 15:14
  • $\begingroup$ Regarding applications of topos theory, what about the classical examples such as crystalline cohomology or elucidating realizability and studying higher-order computability via the effective topos? For a recent and minor example, what about the proof of Grothendieck's generic freeness theorem using topos-theoretic methods (Section 3.5 in the linked notes)? Perhaps you can clarify your questions a bit so we can give more directed answers :-) $\endgroup$ Jan 21 at 18:02

1 Answer 1

14
$\begingroup$

A good answer to both questions is provided by the following variant of the Gelfand duality for commutative von Neumann algebras, which shows that the following categories are equivalent:

  • The category CSLEMS of compact strictly localizable enhanced measurable spaces;

  • The category HStonean of hyperstonean spaces and open maps.

  • The category HStoneanLoc of hyperstonean locales and open maps.

  • The category MLoc of measurable locales, defined as the full subcategory of the category of locales consisting of complete Boolean algebras that admit sufficiently many continuous valuations.

  • The opposite category CVNA^op of commutative von Neumann algebras, whose morphisms are normal *-homomorphisms of algebras in the opposite direction.

The first category, despite the rather complicated name, is essentially the correct category for measure theory: it incorporates equality almost everywhere, a (generalized) σ-finiteness property, and an abstract variant of the Radon measure property, which eliminate pathological measurable (and measure) spaces for which some of the most basic theorem of measure theory (such as the Riesz representation theorem or the Radon–Nikodym theorem) fail.

Of particular interest is the fourth category MLoc of mesurable locales. It is a full subcategory of the category of locales, which quite interesting: it demonstrates that both point-set general topology (as implemented by the category of topological spaces) and point-set measure theory (as implemented by the above category CSLEMS) are a part of pointfree general topology, implemented by full subcategory of the category of locales.

These parts (i.e., general topology and measure theory) are almost disjoint: locales corresponding to topological spaces are spatial, i.e., have enough points. On the other hand, points in a measurable locale are in a bijective correspondence with atoms in the original measure space. In particular, atomless measure spaces (i.e., what is typically used in practice) correspond to locales that have no points at all.

Returning to topos theory: working in the topos of sheaves of sets on a measurable locale amounts to doing ordinary mathematics in measurable families over a measurable space. For example, doing internal linear algebra in such a topos corresponds to working with measurable vector bundles, etc.

$\endgroup$
8
  • $\begingroup$ Great answer! (1) I'm convinced that localizable enhanced measurable spaces are great (by Irving Segal's theorem). Besides the "Gelfund duality theorem" (which is also great), are there other reasons why we should go for CSLEMS? (2) Now I understand that one can almost embed point-set topology and point-set measure theory (provided an answer to 1) into locale theory. Are there essentially other kinds of locale? I mean, given a locale with no (enough, "mixed" resp.) point, how far is it from being a measurable (spatial, "mixed" resp.) locale? $\endgroup$
    – Student
    Jan 21 at 11:02
  • $\begingroup$ (3) The last point you made is also very interesting: Could you point to specific examples where people apply this fact to the old-style topology or algebraic geometry (namely, results not just internal to locale-theorists)? I've also heard that you can do cohomology once you have a good enough locale. Does that apply to CSLEMS? And if so, what kind of cohomology (with concrete examples?) does that provide? $\endgroup$
    – Student
    Jan 21 at 11:04
  • $\begingroup$ @Student: The first question is answered in arXiv:2005.05284. In LEMS one has truly pathological measurable maps (see Remark 5.12), which do not induce a complete homomorphism of Boolean algebras, and are otherwise ill-behaved, e.g., in the context of topos theory we expect a measurable map f:X→Y to induce a Y-indexed measurable family of measurable spaces given by fibers of f, which, however, is simply false for pathological maps like in Remark 5.12. $\endgroup$ Jan 21 at 18:28
  • 1
    $\begingroup$ @Student: Additionally, even if we impose the condition that morphisms in LEMS must induce complete homomorphisms of Boolean algebras by hand, the resulting category has morphisms that look like invertible morphisms for all practical purposes (e.g., they induce an isomorphism of the respective von Neumann algebras and of complete Boolean algebras), but are not actually invertible at the point-set level. The condition of strict localizability and compactness is both necessary and sufficient to eliminate such pathologies, see the remarks before Proposition 5.16. $\endgroup$ Jan 21 at 18:31
  • 1
    $\begingroup$ @Student: This is confirmed by other facts, e.g., by Maharam's theorem we know that measurable spaces and maps of measurable spaces have a very simple structure and admit a complete classification up to an isomorphism. In particular, there is no room for a nontrivial global structure. Likewise, there are no nontrivial vector bundles, fiber bundles, bundle gerbes, or other structures on a measurable space. Topos theory is still interesting in this setting (e.g., you do get to generalize traditional theorems to measurable families), but the topological aspect is trivial. $\endgroup$ Jan 21 at 18:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.