Papers that are (intended to be) submitted to a journal.

Preprint, December 2023.arXiv • Expand abstractCollapse abstract

The classifying topos of a geometric theory is a topos such that geometric morphisms into it correspond to models of that theory. We study classifying toposes for different infinitary logics: first-order, sub-first-order (i.e. geometric logic plus implication) and classical. For the first two, the corresponding notion of classifying topos is given by restricting the use of geometric morphisms to open and sub-open geometric morphisms respectively. For the last one, we restrict ourselves to Boolean toposes instead. Butz and Johnstone proved that the first-order classifying topos of an infinitary first-order theory exists precisely when that theory does not have too many infinitary first-order formulas, up to intuitionistically provable equivalence. We prove similar statements for the sub-first-order and Boolean case. Along the way we obtain completeness results of infinitary sub-first-order logic and infinitary classical logic with respect to (Boolean) toposes.

With Jiri Rosický, preprint, October 2023.arXiv • Expand abstractCollapse abstract

We give a category-theoretic construction of simple and NSOP_{1}-like independence relations in locally finitely presentable categories, and in the more general locally finitely multipresentable categories. We do so by identifying properties of a class of monomorphisms $\mathcal{M}$ such that the pullback squares consisting of morphisms in $\mathcal{M}$ form the desired independence relation. This generalizes the category-theoretic construction of stable independence relations using effective unions or cellular squares by M. Lieberman, S. Vasey and the second author to the unstable setting.

Preprint, May 2023.arXiv • Expand abstractCollapse abstract

We generalise various theorems for finding indiscernible trees and arrays to positive logic: based on an existing modelling theorem for s-trees, we prove modelling theorems for str-trees, str_{0}-trees (the reduct of str-trees that forgets the length comparison relation) and arrays. In doing so we prove stronger versions for basing—rather than locally basing or EM-basing—str-trees on s-trees and str_{0}-trees on str-trees. As an application we show that a thick positive theory has $k$-TP_{2} iff it has 2-TP_{2}.

Papers that have been published or are accepted.

With Anna Dmitrieva and Francesco Gallinaro, accepted in

We give definitions of the properties OP, IP, $k$-TP, TP_{1}, $k$-TP_{2}, SOP_{1}, SOP_{2} and SOP_{3} in positive logic, and prove various implications and equivalences between them. We also provide a characterisation of stability in positive logic in analogy with the one in full first-order logic, both on the level of formulas and on the level of theories. For simple theories there are the classically equivalent definitions of not having TP and dividing having local character, which we prove to be equivalent in positive logic as well. Finally, we show that a thick theory $T$ has OP iff it has IP or SOP_{1} and that $T$ has TP iff it has SOP_{1} or TP_{2}, analogous to the well-known results in full first-order logic where SOP_{1} is replaced by SOP in the former and by TP_{1} in the latter. Our proofs of these final two theorems are new and make use of Kim-independence.

With Vahagn Aslanyan, Robert Henderson and Jonathan Kirby,

We give four different independence relations on any exponential field. Each is a canonical independence relation on a suitable Abstract Elementary Class of exponential fields, showing that two of these are NSOP_{1}-like and non-simple, a third is stable, and the fourth is the quasiminimal pregeometry of Zilber's exponential fields, previously known to be stable (and uncountably categorical). We also characterise the fourth independence relation in terms of the third, strong independence.

We study the model theory of vector spaces with a bilinear form over a fixed field. For finite fields this can be, and has been, done in the classical framework of full first-order logic. For infinite fields we need different logical frameworks. First we take a category-theoretic approach, which requires very little set-up. We show that linear independence forms a simple unstable independence relation. With some more work we then show that we can also work in the framework of positive logic, which is much more powerful than the category-theoretic approach and much closer to the classical framework of full first-order logic. We fully characterise the existentially closed models of the arising positive theory. Using the independence relation from before we conclude that the theory is simple unstable, in the sense that dividing has local character but there are many distinct types. We also provide positive version of what is commonly known as the Ryll-Nardzewski theorem for $\omega$-categorical theories in full first-order logic, from which we conclude that bilinear spaces over a countable field are $\omega$-categorical.

The classes stable, simple and NSOP_{1} in the stability hierarchy for first-order theories can be characterised by the existence of a certain independence relation. For each of them there is a canonicity theorem: there can be at most one nice independence relation. Independence in stable and simple first-order theories must come from forking and dividing (which then coincide), and for NSOP_{1} theories it must come from Kim-dividing.

We generalise this work to the framework of AECats (Abstract Elementary Categories) with the amalgamation property. These are a certain kind of accessible category generalising the category of (subsets of) models of some theory. We prove canonicity theorems for stable, simple and NSOP_{1}-like independence relations. The stable and simple cases have been done before in slightly different setups, but we provide them here as well so that we can recover part of the original stability hierarchy. We also provide abstract definitions for each of these independence relations as what we call isi-dividing, isi-forking and long Kim-dividing.

With Jan Dobrowolski,

An important dividing line in the class of unstable theories is being NSOP_{1}, which is more general than being simple. In NSOP_{1} theories forking independence may not be as well-behaved as in stable or simple theories, so it is replaced by another independence notion, called Kim-independence. We generalise Kim-independence over models in NSOP_{1} theories to positive logic—a proper generalisation of first-order logic where negation is not built in, but can be added as desired. For example, an important application is that we can add hyperimaginary sorts to a positive theory to get another positive theory, preserving NSOP_{1} and various other properties. We prove that, in a thick positive NSOP_{1} theory, Kim-independence over existentially closed models has all the nice properties that it is known to have in a first-order NSOP_{1} theory. We also provide a Kim-Pillay style theorem, characterising which thick positive theories are NSOP_{1} by the existence of a certain independence relation. Furthermore, this independence relation must then be the same as Kim-independence. Thickness is the mild assumption that being an indiscernible sequence is type-definable.

In first-order logic Kim-independence is defined in terms of Morley sequences in global invariant types. These may not exist in thick positive theories. We solve this by working with Morley sequences in global Lascar-invariant types, which do exist in thick positive theories. We also simplify certain tree constructions that were used in the study of Kim-independence in first-order theories. In particular, we only work with trees of finite height.

**Corrigendum** (January 2024). The original proof of the independence theorem contains a gap. The theorem is still true, and in the corrigendum we give a different proof.

We construct a 2-equivalence $\mathfrak{CohTheory}^\text{op} \simeq \mathfrak{TypeSpaceFunc}$. Here $\mathfrak{CohTheory}$ is the 2-category of positive theories and $\mathfrak{TypeSpaceFunc}$ is the 2-category of type space functors. We give a precise definition of interpretations for positive logic, which will be the 1-cells in $\mathfrak{CohTheory}$. The 2-cells are definable homomorphisms. The 2-equivalence restricts to a duality of categories, making precise the philosophy that a theory is 'the same' as the collection of its type spaces (i.e. its type space functor).

In characterising those functors that arise as type space functors, we find that they are specific instances of (coherent) hyperdoctrines. This connects two different schools of thought on the logical structure of a theory.

The key ingredient, the Deligne completeness theorem, arises from topos theory, where positive theories have been studied under the name of coherent theories.

We introduce the framework of AECats (abstract elementary categories), generalising both the category of models of some first-order theory and the category of subsets of models. Any AEC and any compact abstract theory ("cat", as introduced by Ben-Yaacov) forms an AECat. In particular, we find applications in positive logic and continuous logic: the category of (subsets of) models of a positive or continuous theory is an AECat.

The Kim-Pillay theorem for first-order logic characterises simple theories by the properties dividing independence has. We prove a version of the Kim-Pillay theorem for AECats with the amalgamation property, generalising the first-order version and existing versions for positive logic.

Talks and the corresponding material.

Talk at Oxford Logic Seminar, February 2024.Slides (PDF) • Expand abstractCollapse abstract

Positive logic is a generalisation of full first-order logic, where negation is not built in, but can be added as desired. In joint work with Jan Dobrowolski we succesfully generalised the recent development on Kim-independence in NSOP_{1} theories to the positive setting. One of the important theorems in this development is the independence theorem, whose statement is very similar to the well-known statement for simple theories, and allows us to amalgamate independent types. In this talk we will have a closer look at the proof of this theorem, and what needs to be changed to make the proof work in positive logic compared to full first-order logic.

Talk at Model Theory Conference in Seoul, August 2023.Slides (PDF) • Expand abstractCollapse abstract

This is joint work with Francesco Gallinaro and Anna Dmitrieva. We give definitions of the properties OP, IP, *k*-TP, TP_{1}, *k*-TP_{2}, SOP_{1}, SOP_{2} and SOP_{3} in positive logic, a proper generalisation of full first-order logic where negation is not built in, but can be added as desired. We prove various implications and equivalences between these properties, matching those known from full first-order logic. We will look at an example illustrating why these definitions have to be altered for positive logic. We will also give positive versions of the well-known results that a theory is stable iff it is NSOP and NIP, and that a theory is simple iff it is NTP_{1} and NTP_{2}. In the positive versions of these theorems we replace NSOP and NTP1 respectively by NSOP_{1}. This allows us to give completely new proofs based on Kim-independence.

Talk at SEEMOD XIV in Buckingham, March 2023, also given at One-day of Algebra and Model Theory in Manchester, January 2023.Slides (PDF) • Expand abstractCollapse abstract

Neostability focuses on developing the powerful tools and ideas for stable theories in broader, less well-behaved, classes of theories. This has been hugely successful in simple theories and more recently in NSOP_{1} theories. A central notion in this context is that of an independence relation, which in stable and simple theories comes from forking and in NSOP_{1} theories it comes from Kim-forking. There are many mathematically interesting structures that cannot be studied in the classical framework of first-order logic, but where ideas from neostability, such as independence, apply. For this reason, neostability has been developed in more general logical frameworks, such as continuous logic, positive logic and a very general category-theoretic approach that subsumes all these frameworks. In this talk I will focus on the categorical approach to independence relations, and how even in this general setting we get canonicity theorems that are surprisingly close to those we know from first-order logic. I will finish with recent examples of where the categorical framework can be applied.

Talk at ECIMT 2022, also given at UEA Logic seminar, QMUL and Manchester logic seminar, March 2022.Slides (PDF) • Expand abstractCollapse abstract

We study the model theory of vector spaces with a bilinear form over a fixed field. For finite fields this can, and has been, done in the classical framework of full first-order logic. For infinite fields we run into issues with compactness and we need a different logical framework. In this talk we will take the approach of positive logic, a framework that is very close to full first-order logic, but where negation is not built in (but can be added as desired). Positive logic allows us to study bilinear spaces over any fixed field. The arising positive theory turns out to be simple unstable. We also fully characterise its existentially closed models. Time permitting, we will also discuss some other interesting facts about this theory, concerning elimination of quantifiers and omega-stability.

Talk at the British Postgraduate Model Theory Conference (BPGMTC) 2022, January 2022.Slides (PDF) • Expand abstractCollapse abstract

Independence relations play a central role in model theory. We all already know examples of independence relations, for example linear independence in vector spaces. Shelah generalised the idea of independence, through the notion of forking, to a very broad class of mathematical structures: the so-called stable structures. In vector spaces (which are stable structures) this forking independence then coincides with linear independence. This is no coincidence, because it turns out that independence is canonical. That is, in any structure there is at most one nice enough independence relation, and if it exists it coincides with forking independence. Later, Kim and Pillay generalised work on forking independence to the even broader class of simple structures. This does come at the cost of a certain property that forking has in stable structures, but we keep enough properties for a canonicity theorem. Going one step further there is the class of NSOP_{1} structures, which is more general than the class of simple structures. Kaplan and Ramsey were able to generalise the work on independence to this NSOP_{1} class. This comes at the cost of another property, but we still keep canonicity (technical note: we now have to replace forking by Kim-forking).

All of the above takes place in the framework of first-order logic. Many interesting mathematical structures do not fit well in this framework. In this talk we will have a look at results that generalise these theorems on canonicity of independence to a very general category-theoretic framework, called Abstract Elementary Categories (AECats). That is, after translating the properties that we know independence has in stable / simple / NSOP_{1} structures to our category-theoretic framework, we get a definition of a stable / simple / NSOP_{1}-like independence relation. The main results then state that there can be at most one stable / simple / NSOP_{1}-like independence relation on an AECat and that it has to be given by an appropriate generalisation of forking (or Kim-forking for NSOP_{1}-like independence).

Talk at the UEA pure mathematics seminar, October 2021.Slides (PDF) • Expand abstractCollapse abstract

Independence relations play a central role in model theory. We all already know examples of independence relations, for example linear independence in vector spaces. Shelah generalised the idea of independence, through the notion of forking, to a very broad class of mathematical structures: the so-called stable structures. In vector spaces (which are stable structures) this forking independence then coincides with linear independence. This is no coincidence, because it turns out that independence is canonical. That is, in any structure there is at most one nice enough independence relation, and if it exists it coincides with forking independence. Later, Kim and Pillay generalised work on forking independence to the even broader class of simple structures. This does come at the cost of a certain property that forking has in stable structures, but we keep enough properties for a canonicity theorem. Going one step further there is the class of NSOP_{1} structures, which is more general than the class of simple structures. Kaplan and Ramsey were able to generalise the work on independence to this NSOP_{1} class. This comes at the cost of another property, but we still keep canonicity (technical note: we now have to replace forking by Kim-forking).

All of the above takes place in the framework of first-order logic. Many interesting mathematical structures do not fit well in this framework. In this talk we will have a look at results that generalise these theorems on canonicity of independence to a very general category-theoretic framework, called Abstract Elementary Categories (AECats). That is, after translating the properties that we know independence has in stable / simple / NSOP_{1} structures to our category-theoretic framework, we get a definition of a stable / simple / NSOP_{1}-like independence relation. The main results then state that there can be at most one stable / simple / NSOP_{1}-like independence relation on an AECat and that it has to be given by an appropriate generalisation of forking (or Kim-forking for NSOP_{1}-like independence).

Talk at the Helsinki Logic Seminar, September 2021.Slides (PDF) • Expand abstractCollapse abstract

In Shelah's stability hierarchy we classify theories using combinatorial properties. Some important classes are: stable, simple and NSOP_{1}, each being contained in the next. We can characterise these classes by the existence of a certain independence relation. For example, in vector spaces such an independence relation comes from linear independence. Part of this characterisation is canonicity of the independence relation: there can be at most one nice enough independence relation in a theory.

Lieberman, Rosický and Vasey proved canonicity of stable-like independence relations in accessible categories. Accessible categories are a very general framework. The category of models of some theory is an accessible category, every AEC (abstract elementary class) is an accessible category, but even then accessible categories are more general. Inspired by this we introduce the framework of AECats (abstract elementary categories) and prove canonicity for simple-like and NSOP_{1}-like independence relations. This way we reconstruct part of the hierarchy that we have for first-order theories, but now in the very general category-theoretic setting.

Talk at SMTH2, June 2021.Slides (PDF) • Expand abstractCollapse abstract

This is joint work with Jan Dobrowolski. Positive logic is a proper generalisation of first-order logic where negation is not built in, but can be added as desired. In this talk I will give a brief introduction to positive logic. We will have a look at the challenges that positive logic presents us and how they can be solved. I will also explain why positive logic is useful and how we can use it to study structures that do not fit the usual framework of first-order logic.

An important dividing line in the class of unstable theories is being NSOP1, which is more general than being simple. There has been a lot of recent work on the class of NSOP1 theories for first-order logic. The natural independence relation in this class is given by Kim-independence, generalising forking independence from stable and simple theories. We have generalised work on Kim-independence to the setting of positive logic. I will assume no prior knowledge of this topic and will introduce the definitions of NSOP1 and Kim-independence, and how we can make sense of this in positive logic. Our results can be summarised as a Kim-Pillay style theorem: a thick positive theory is NSOP1 if and only if there is a nice enough independence relation, and in this case the independence relation is given by Kim-independence.

Talk at the Masaryk University Algebra Seminar, April 2021.Slides (PDF) • Youtube • Expand abstractCollapse abstract

In Shelah's classification of first-order theories we classify theories using combinatorial properties. The most well-known class is that of stable theories, which are very well-behaved. Simple theories are more general, and then even more general are the NSOP_{1} theories. We can characterise those classes by the existence of a certain independence relation. For example, in vector spaces such an independence relation comes from linear independence. Part of this characterisation is canonicity of the independence relation: there can be at most one nice enough independence relation in a theory.

Lieberman, Rosický and Vasey proved canonicity of stable-like independence relations in accessible categories. Inspired by this we introduce the framework of AECats (abstract elementary categories) and prove canonicity for simple-like and NSOP_{1}-like independence relations. This way we reconstruct part of the hierarchy that we have for first-order theories, but now in the very general category-theoretic setting.

Talk at GSCL XXII, March 2021.Slides (PDF) • Expand abstractCollapse abstract

This is joint work (in progress) with Jan Dobrowolski. Positive logic is a proper generalisation of first-order logic where we only allow the logical connectives for conjunction, disjunction, falsehood, truth and the existential quantifier. For example, if we add hyperimaginaries as parameters to our monster model then we leave the framework of first-order logic, but we remain in the framework of positive logic. Another interesting example is the positive theory of existentially closed exponential fields (ECEF), introduced by Haykazyan and Kirby.

There has been a lot of recent work on the class of NSOP1 theories for first-order logic. The natural independence relation in this class is given by Kim-independence. We have generalised the work on Kim-independence to the setting of positive logic. In this talk I will explain the challenges that positive logic presents us and how we solve them. Our result can then be summarised as a Kim-Pillay style theorem: a thick positive theory is NSOP1 if and only if there is a nice enough independence relation, and in this case the independence relation is given by Kim-dividing.

Haykazyan and Kirby proved that ECEF has a nice enough independence relation and showed that it is thus NSOP1. So our theorem applies and confirms that this independence relation is given by Kim-dividing. The operation of adding hyperimaginaries as parameters preserves the NSOP1 property. So in particular, if we start with an NSOP1 first-order theory and we add hyperimaginaries then our theorem still applies.

Talk at the logic seminar at the Hebrew University of Jerusalem, December 2020.Slides (PDF) • Expand abstractCollapse abstract

This is joint work (in progress) with Jan Dobrowolski. Positive logic is a proper generalisation of first-order logic where we only allow the logical connectives for conjunction, disjunction, falsehood, truth and the existential quantifier. For example, if we add hyperimaginaries as parameters to our monster model then we leave the framework of first-order logic, but we remain in the framework of positive logic. Another interesting example is the positive theory of existentially closed exponential fields (ECEF), introduced by Haykazyan and Kirby.

There has been a lot of recent work on the class of NSOP1 theories for first-order logic. The natural independence relation in this class is given by Kim-independence. We have generalised the work on Kim-independence to the setting of positive logic. In this talk I will explain the challenges that positive logic presents us and how we solve them. Our result can then be summarised as a Kim-Pillay style theorem: a thick positive theory is NSOP1 if and only if there is a nice enough independence relation, and in this case the independence relation is given by Kim-dividing.

Haykazyan and Kirby proved that ECEF has a nice enough independence relation and showed that it is thus NSOP1. So our theorem applies and confirms that this independence relation is given by Kim-dividing. The operation of adding hyperimaginaries as parameters preserves the NSOP1 property. So in particular, if we start with an NSOP1 first-order theory and we add hyperimaginaries then our theorem still applies.

Poster for the Topics in Category Theory 2020 spring school, March 2020.Poster (PDF) • Expand abstractCollapse abstract

For a first-order theory $T$ we can collect the type spaces $\operatorname{S}_n(T)$ in a contravariant functor between the category of finite sets and the category of Stone spaces. Haykazyan generalised this idea to positive theories, replacing Stone spaces by spectral spaces. We characterise those functors that arise as a positive type space functor. This results in a duality between the category of positive theories with strong interpretations and the category of type space functors. Using the Stone duality between spectral spaces and distributive lattices, we can alternatively view type space functors as functors into the category of distributive lattices. This gives another characterisation of the same functors, namely as specific instances of coherent hyperdoctrines. The key ingredient, the Deligne completeness theorem, arises from topos theory, where positive theories have been studied under the name of coherent theories.

Miscellaneous notes and pieces of mathematics.

Short note translating some standard tools in model theory to AECats. PDF • Expand abstractCollapse abstract

In this note we translate standard tools and arguments in model theory to the framework of finitely short AECats. In particular we prove that a form of compactness holds. From this we then get some standard tools for manipulating and building indiscernible sequences.

My theses.

PhD thesis about model-theoretic independence relations in AECats (Abstract Elementary Categories) and positive logic. Supervised by Jonathan Kirby, at University of East Anglia.PDF • More info • Expand abstractCollapse abstract

In model theory, a branch of mathematical logic, we can classify mathematical structures based on their logical complexity. This yields the so-called stability hierarchy. Independence relations play an important role in this stability hierarchy. An independence relation tells us which subsets of a structure contain information about each other, for example: linear independence in vector spaces yields such a relation.

Some important classes in the stability hierarchy are stable, simple and NSOP_{1}, each being contained in the next. For each of these classes there exists a so-called Kim-Pillay style theorem. Such a theorem describes the interaction between independence relations and the stability hierarchy. For example: simplicity is equivalent to admitting a certain independence relation, which must then be unique.

All of the above classically takes place in full first-order logic. Parts of it have already been generalised to other frameworks, such as continuous logic, positive logic and even a very general category-theoretic framework. In this thesis we continue this work.

We introduce the framework of AECats, which are a specific kind of accessible category. We prove that there can be at most one stable, simple or NSOP_{1}-like independence relation in an AECat. We thus recover (part of) the original stability hierarchy. For this we introduce the notions of long dividing, isi-dividing and long Kim-dividing, which are based on the classical notions of dividing and Kim-dividing but are such that they work well without compactness.

Switching frameworks, we generalise Kim-dividing in NSOP_{1} theories to positive logic. We prove that Kim-dividing over existentially closed models has all the nice properties that it is known to have in full first-order logic. We also provide a full Kim-Pillay style theorem: a positive theory is NSOP_{1} if and only if there is a nice enough independence relation, which then must be given by Kim-dividing.

Master's thesis about classifying topoi and connection to model theory. Supervised by Jaap van Oosten, at Utrecht University.PDF • Expand abstractCollapse abstract

Every geometric theory has a classifying topos, but when trying to extend this to full first-order theories one may run into trouble. Such a first-order classifying topos for a first-order theory $T$, is a topos $\mathcal{F}$ such that for every topos $\mathcal{E}$, the models of $T$ in $\mathcal{E}$ correspond to open geometric morphisms $\mathcal{E} \to \mathcal{F}$. The trouble is that not every first-order theory may have such a first-order classifying topos, as was pointed out by Carsten Butz and Peter Johnstone. They characterized which theories do admit such a first-order classifying topos, and show how to construct such a first-order classifying topos.

The work of Butz and Johnstone is the main subject of this thesis. The construction of a classifying topos for both geometric theories and first-order theories is worked out in detail. We will also study the characterization of which theories admit a first-order classifying topos. In doing so, we obtain certain completeness results that are interesting in their own right. These are completeness results for deduction-systems for various kinds of infinitary logic, with respect to models in topoi. Building on top of those results, we also obtain a completeness result for classical infinitary logic, with respect to Boolean topoi.

One of the first goals of this thesis was to form a link between Topos Theory and Model Theory, via the first-order classifying topos. In order to bridge the gap between the intuitionistic logic of topoi and the classical logic of Model Theory, we introduce the concept of a Boolean classifying topos. We provide a characterization of which first-order theories admit such a Boolean classifying topos, much like the one for first-order classifying topoi. Then we give a simple example of how to link Boolean classifying topoi to Model Theory, by characterizing complete theories in terms of their Boolean classifying topos.

Bachelor's thesis about Gödel's incompleteness theorems. Supervised by Jaap van Oosten, at Utrecht University.PDF • Expand abstractCollapse abstract

The first incompleteness theorem of Kurt Gödel states that a theory in which we can develop most of modern arithmetic is incomplete. We will take a look at such a theory: Peano Arithmetic (PA). We will develop the tools to formulate a sentence that essentially asserts in PA that it is not provable. Then we use this sentence to prove both Gödel's first and second incompleteness theorems, where the second states that PA cannot prove its own consistency. To prove Gödel's first incompleteness theorem also for consistent extensions of PA we will use Rosser sentences. These are sentences that are equivalent in PA to the assertion that a disproof of them occurs before any proof of them. After developing the necessary technical tools, we prove that Rosser sentences need not to be all provably equivalent but there are constructions where they are *(after a paper by Guaspari and Solovay)*.

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