Spring 2025

We will introduce the notion of a coz-included sublocale. It is a necessary condition for a sublocale to be a dense C-embedding, and together with the property of being well-embedded, is sufficient. However, although it is clearly a strengthening of coz-ontoness (or z-embeddings), it is incomparable with C*-embeddings as is demonstrated by the Cech-Stone compactification of the real numbers.  

In this talk, taking advantage of the algebraic nature of frames, we will first motivate this notion from a lattice-theoretic perspective. Then, using the geometric behavior of the category of locales (the dual category of frames),  we will give a topological interpretation of this notion.  We will provide non-trivial spatial examples of this new notion and other weakenings, such as casi coz-included and relative zero sublocales, and explore and characterize them in the point-free setting.  If time allows, we will present new characterizations of several classes of frames, such as perfectly normal and Oz frames, among others.

Categories are simple structures whose definition, and fundamental theory, is well known and universally agreed upon. Consequently, they have become a fundamental organisational tool in a broad swathe of mathematical disciplines.

Their higher generalisations, n-categories and ∞-categories, however, are not simple to define and use nor do they even enjoy a unitary definition. Instead, we validate our ∞-categorical intuition by building complex homotopical models, which are often bespoke to a particular application domain. This private zoo of exotic species has grown rapidly over the past 2 decades, with a far from non-exhaustive list of its inhabitants including the quasi-categories, (complete) Segal spaces, Joyal-Rezk 𝜃-sets and spaces, Street-Verity complicial (and indeed comical) sets, Gray-categories, Tamsamani-Simpson n-categories, Batanin-Leinster n-categories, Baez-Dolan opetopic n-categories, Trimble n-categories and so on and on…

In many cases we know that these models enjoy equivalent homotopy theories, though in others we may not even know that, and this alone does not ensure that we may easily translate category theoretic notions from one to the next. The implementations of those are often tied inextricably to the specific set theoretic bits-and-bobs from which a given model is constructed and, indeed, in some cases that detail may obscure the very meaning and intent of a given categorical construction.

The thesis of this talk is that we may begin to peer through this morass and glimpse a nirvana of ∞-categories for all by consciously decoupling these analytic models from the synthetic languages we use to describe the categorical constructions they support. In the process, we shall see that the distinction between analytic and synthetic approaches is not a binary one, instead it is a continuum parameterised by decisions we make about what to capture in the languages we construct and what to elide.

Along the way, I’d like also to illustrate some of the beautiful geometry that lurks in the foundations of higher category theory itself.

In this talk I will present some recent results concerning the amalgamation property (AP) in varieties (equational classes) of residuated lattices. Varieties of (pointed) residuated lattices provide algebraic semantics for substructural logics.  A variety of residuated lattices having the AP amounts to its corresponding substructural logic enjoying certain interpolation properties. This provides another motivation to investigate the AP for varieties of residuated lattices in addition to the purely abstract algebraic one.

In this talk, we investigate connections between the family of comparative logics including Abelian logic, and some generalizations of Lukasiewicz logic.

Lukasiewicz logic in its infinitely-valued version was introduced by Lukasiewicz and Tarski in 1930 and since then it proved to be one of the most prominent non-classical logics. 

Abelian logic is a well-known contraclassical paraconsistent logic. This logic was independently introduced by Meyer and Slaney and by Casari and it is also called the logic of Abelian ℓ-groups or Abelian Group Logic. This terminology follows from the fact that the matrix models of Abelian logic consist of Abelian ℓ-groups and their positive cones as filters of designated elements.

Łukasiewicz unbound logic was introduced as a generalization of Łukasiewicz logic. In addition to philosophical and linguistic motivations, this logic can also be motivated purely syntactically, in that the connectives of the unbound Łukasiewicz logic can be seen as an untruncated version of the connectives of the standard Łukasiewicz logic.

In this talk we are interested in logics which are semilinear, i.e. they are determined by their totally ordered models. By additional axioms, we may force realizations of 0 and f to appear in a specific order. For example, we show that the models of pointed Abelian logic where f is interpreted as strictly smaller then 0 corresponds to the models of unbound Lukasiewicz logic. We provide axiomatizations for these logics and we discuss their finite strong completeness properties and their possible philosophical interpretations.

This talk will survey some of my recent results, both published and unpublished, on (un)decidability within modal and temporal logic, truthmaker semantics, team semantics, and relevant logic (to the extent possible, during the talk, I will also phrase results algebraically). Motivated by the question of what makes a logic (or equational theory) undecidable, the aim is to identify traits of undecidable logics. As a consequence, the presentation will offer more breadth than depth.

To begin, we show that the modal logic of any class of semilattices containing (P(N), U) is undecidable. (As a fun aside, this result was actually obtained during my previous visit to Chapman in November 2023, so it feels fitting to present it here now.) The result carries substantial implications, including resolving the open problem concerning the decidability of hyperboolean modal logic, as raised by Goranko and Vakarelov (1999).

Next, we compare this result with the decidability of team, truthmaker, and modal information logics, drawing heuristic insights into the factors contributing to the decidability or undecidability of logics.

Finally, we turn our attention to the semilattice relevant logic S, introduced by Urquhart (1972). While Urquhart (1984) showed that many relevant logics are undecidable, S eluded these techniques. Eventually, it led Urquhart (2016) to conjecture that S is decidable. Contrary to this conjecture, we show that S is undecidable.

References

• Goranko, Valentin and Dimiter Vakarelov (1999). “Hyperboolean Algebras and Hyperboolean Modal Logic”. In: Journal of Applied Non-Classical Logics 9.2-3, pp. 345–368.

• Urquhart, Alasdair (1972). “Semantics for relevant logics”. In: Journal of Symbolic Logic 37, pp. 159–169.

• Urquhart, Alasdair (1984). “The undecidability of entailment and relevant implication”. In: Journal of Symbolic Logic 49, pp. 1059–1073.

• Urquhart, Alasdair (2016). “Relevance Logic: Problems Open and Closed”. In: The Australasian Journal of Logic 13.

In contrast to ordinary categories, ∞-categories can be described as categories "enriched in spaces", meaning they consist of a space of objects, spaces of morphisms, and continuous composition maps. This exhibits them as topological or homotopical objects, so they should be amenable to being captured within a synthetic, homotopical framework such as homotopy type theory (HoTT)---a constructive foundation system in which homotopy equivalent types are perceived as equal, as per Voevodsky's univalence axiom.

It is not known how to achieve this in basic HoTT, but there exists an extension due to Riehl--Shulman in which one adds an interval witnessing directed arrows in a type. It turns out that this unlocks the development of a fair amount of ∞-category theory in this synthetic setting.

However, it has been a long-standing defect of this theory that it seemingly lacks the power to construct actual examples of nontrivial ∞-categories.

I present recent work in which we are adding certain modal operators to the theory (such as the opposite type, the maximal subgroupoid, and the twisted arrow type) making it possible to define the ∞-category of spaces aka ∞-groupoids. We can furthermore prove that it is complete and cocomplete, and directed univalent, i.e., geometric arrows in the universe are functors between ∞-groupoids.

As an application, this allows us to derive from it various other categories of (higher) algebraic structures such as the category of posets, reflexive graphs, the simplex category, the category of homotopy-coherent monoids and groups, resp., and the category of spectra.

As another application, we are able to prove the ∞-version of the classical Yoneda lemma, drastically simplifying the steps that would be necessary in ordinary set-theoretic foundations to prove functoriality. This also enables the development of ∞-presheaf theory in that setting.

If time permits, as a final application, I will mention some first major results about Kan extensions and cofinal functors, including a new, synthetic proof of Quillen's Theorem A, a celebrated result in the homotopy theory of categories.