# Accessible categories and their connections

## - set theory - model theory - homotopy theory -

### 17-20 July 2018 — School of Mathematics, University of Leeds

All talks will take place in the MALL, 8th floor, School of Mathematics, University of Leeds. Coffee breaks will almost all take place on the 9th floor.

Tuesday 17thWednesday 18thThursday 19thFriday 20th
09:30-10:009:45-10:45
Registration and coffee
10:45-11:00
Welcome (MALL)
Victoria GitmanJavier GutierrezPedro Zambrano
10:00-10:30
10:30-11:00Coffee breakCoffee breakCoffee break
11:00-11:30Tibor BekeConstanze RoitzheimJiri RosickyJohn Bourke
11:30-12:00
12:00-12:30Magdalena KedziorekStamatis DimopoulosJoan BagariaAndres Villaveces
12:30-13:00
13:00-13:30Lunch
13:30-14:00
14:00-14:30
14:30-15:00Mike LiebermanShoshana FriedmanWill BoneyDiscussion
15:00-15:30
15:30-16:00Coffee break
16:00-16:30Carles CasacubertaSebastien VaseyMenachem Magidor
16:30-17:00
18:00-???Social Dinner

### Titles & abstracts

#### Joan Bagaria

Some considerations around the Weak Vopenka Principle

#### Tibor Beke

Accessible categories and model theory: hits and misses

Starting with Lawvere's "functorial semantics" in the 1960's, category theory coexisted (and sometimes, competed) with Tarski-style model theory to serve as a language of metamathematics. The 1989 monograph of Makkai and Pare, "Accessible categories: the foundations of categorical model theory" provides, I believe, both the most systematic and most beautiful framework for category theory to serve model theory, and vice versa. I will give a (necessarily partial and subjective) overview of where the ideas of Makkai and Pare have taken us during these three decades, and where they might take us in the future. I will give examples where the language of category theory allows for elegant (re)formulations of theorems of Hodges and Shelah, as well as examples where category theory cannot seem to go on its own. Almost all the material in this talk is based on my collaboration with Jiri Rosicky as well as conversations with Michael Makkai.

#### Will Boney

Erdos-Rado classes; or What categories characterize large, finitely accessible categories by faithfully and colimit-preservingly embedding into them?

We present preliminary results on Erdos-Rado classes. From a combinatorial set theory perspective, these are the classes that generalize the Erdos-Rado Theorem in the same way that Ramsey Classes generalize Ramsey's Theorem. From a model theoretic perspective, these are the classes from which we can build generalized indiscernibles and generalized Ehrenfuecht-Mostowski models in $\mathbb{L}_{\infty, \omega}$-axiomatizable classes. From a category theoretic perspective, these are the (concrete) categories that embed into every large, finitely accessible category via a faithful and colimit-preserving functor. We will discuss examples and some applications.

#### John Bourke

Accessible aspects of 2-category theory

Two-dimensional universal algebra is primarily concerned with categories equipped with structure and functors preserving that structure up to coherent isomorphism. The 2-category of monoidal categories and strong monoidal functors is a good example of this. It has been known since the 1980s that such 2-categories admit all weak 2-categorical colimits, but not strict colimits such as coequalisers. In particular, such 2-categories are not locally presentable in the usual sense. However it turns out, rather surprisingly, that the underlying categories of such 2-categories are often accessible, in the usual sense. These results, which stem from discussions between myself and Michael Makkai, allow us to apply the ordinary theory of accessible categories to establish the key properties of such 2-categories. I will discuss some of these results and, furthermore, conjecture about the accessibility of various categories of weak algebraic structures and weak maps between them.

#### Carles Casacuberta

Preservation of colimits under changes of universe

We aim to study under which conditions limits or colimits are preserved under changes of universe. For this, we first need to make precise what is meant by changing a category from a universe to another universe. Although we expected that accessible categories would behave particularly well for this purpose, we found that accessibility itself need not be preserved under changes of universe. This is joint work in progress with Andrew Brooke-Taylor.

#### Stamatis Dimopoulos

Vopenka's Principle and Woodin-like cardinals

Vopenka's Principle has evolved into one of the strongest connections among category theory, model theory and set theory. In this talk, we will give a brief overview of the historical development of the principle, show some of its numerous characterisations and contrast it with large cardinal notions from set theory, especially with Woodin cardinals. This will naturally lead to the definition of a new type of large cardinal, called "Woodin for strong compactness" and we will discuss some of its properties. No prior knowledge of the theory of large cardinals will be assumed.

#### Shoshana Friedman

Prying HOD and $V$ apart with HOD-supercompactness

Woodin's HOD Conjecture proposes that HOD, the class of hereditarily ordinal definable sets, is close to the mathematical universe $V$ in a particular way. In this talk I will give a brief overview of some of the theorems, motivations and definitions relating to the HOD Conjecture. Then I will describe the ways in which HOD can made be quite different from $V$ via forcing, particularly by separating the concepts of supercompactness, supercompactness in HOD and being HOD-supercompact. Where possible, I hope to show where the concepts of C$^{(n)}$ cardinals can play a role in these implications.

#### Victoria Gitman

Virtual Vopenka's Principle

A major segment of modern research in set theory focuses on the topics of forcing and large cardinals. Forcing, introduced by Cohen to show the independence of the Continuum Hypothesis, is a technique for building set-theoretic universes satisfying a desired list of properties. Large cardinal axioms are strong axioms positing the existence of very large infinite objects and (often) elementary embeddings between set-theoretic structures. These axioms form a hierarchy against which the strength of any set-theoretic assertion can be measured. Forcing and large cardinals interact in a number of unexpected ways. In this talk, I will discuss a relatively new avenue of research involving the notion of virtual large cardinals. Given a large cardinal property A characterized by the existence of elementary embeddings of first-order structures, we say that the property A holds virtually if embeddings on structures from the universe V characterizing A exists in forcing extensions of V . Virtual large cardinals are much weaker than their original counterparts, for instance being compatible with the constructible universe L. This line of research leads naturally to considering a virtual version of Vopenka’s Principle, a large cardinal principle which has found applications in other branches of math. Virtual Vopenka’s Principle states that given a proper class of first-order structures in a common language, two of them must elementary embed in a forcing extension of V . I will discuss the consistency strength of the new principle, which like other virtual large cardinal principles, is compatible with L, and some open questions concerning stronger version of the principle, where we demand that the forcing extension in which the embeddings exist preserves a large initial segment of the universe. Parts of this work are joint with Bagaria, Schindler, and Hamkins.

#### Javier Gutierrez

Postnikov sections of model categories.

Given a Quillen adjunction of two variables $\mathcal{C}\times \mathcal{D}\to \mathcal{E}$ between combinatorial model categories and a set of morphisms $\mathcal{S}$ in $\mathcal{C}$, we will prove that there is a localized model structure $L_\mathcal{S}\mathcal{E}$ on $\mathcal{E}$, where the local objects are described via the right adjoint. This new model structure generalizes (enriched) left Bousfield localizations and Barnes and Roitzheim's familiar model structures. We will show how to use this approach to build Postnikov sections on arbitrary combinatorial model categories. This is a joint work with C. Roitzheim.

#### Magdalena Kedziorek

Accessible model categories

In this talk I will discuss the importance of accessible categories in abstract homotopy theory. We will see how to create new model structures from existing ones quite easily, provided we are in the world of accessible categories. I will present parts of joint work with Hess, Riehl and Shipley and also with Garner and Riehl.

#### Mike Lieberman

Extensions of ZF through the lens of accessible categories

We discuss a few recent (and a few less recent) results on the broad consequences of certain familiar set-theoretic hypotheses for the structure of accessible categories. In particular, we consider the reduction of the sharp inequality relation to cardinal arithmetic, and the resulting simplification of the accessibility spectrum (and notion of internal size) under GCH, or even instances of SCH. We also consider several large cardinal assumptions via their clear, and often mathematically natural, implications for (or equivalents in) accessible categories. We propose, in the humblest possible way, that the search for equivalents of large cardinal axioms, in particular, is a program of research eminently worth pursuing. This is joint work with Jiří Rosický and/or Sebastien Vasey.

#### Menachem Magidor

Generalized logics, Vopenka's Principle and accessible categories

#### Constanze Roitzheim

Framings in homotopy theory

One key objective in homotopy theory is finding functors between model categories that respect the homotopy structure, so-called Quillen functors. Framings provide a way to construct and classify Quillen functors from simplicial sets to any given model category. We will then proceed to a more structured set-up where one studies such framings in a stable setting and how it can be used to study the deeper architecture of the stable homotopy category.

#### Jiri Rosicky

Forking in accessible categories

Forking is a central notion of model theory, generalizing linear independence in vector spaces and algebraic independence in fields. We develop the theory of forking in arbitrary accessible categories. To do so, we present an axiomatic definition of what we call a stable independence relation and show that this is in fact a purely category-theoretic axiomatization of the properties of model-theoretic forking in a stable first-order theory. The definition lists properties of commutative squares which are declared to be independent and is primarily suitable for accessible categories whose morphisms are monomorphisms, i.e., to categories appearing in model theory. We show that an accessible category with directed colimits whose morphisms are monomorphisms can have at most one stable independence relation. Its existence has strong consequences, including stability and tameness (in the model-theoretic sense). Any coregular locally presentable category with effective unions has a stable indepen- dence relation for regular monomorphisms consisting of pullback squares. Assuming a large cardinal axiom, an accessible category with directed colimits whose morphisms are monomorphisms has a cofinal full subcategory with a stable independence relation if and only if a certain order property fails.

References: [1] M. Lieberman, J. Rosick ́y and S. Vasey, Forking independence from the categorical point of view, arXiv:1801.09001.

#### Sebastien Vasey

Categoricity and multidimensional diagrams

The categoricity spectrum of a class of structures is the class of cardinals k such that the class has exactly one model of cardinality k up to isomorphism. Shelah's eventual categoricity conjecture is the statement that the categoricity spectrum of an abstract elementary class (AEC) is either bounded or contains an end segment. Roughly, an AEC is a concrete accessible category with directed colimits satisfying a few extra properties. AECs generalize, for example, classes of models of a first-order theory, classes of models in several infinitary logics, and finitely accessible categories with all morphisms monomorphisms. Shelah's eventual categoricity conjecture is one of the main test questions in non-elementary model theory, but despite forty years and many more pages of partial approximations it remains open. Until now, it was not even known to be consistent with large cardinals. In this talk, I will present a joint work with Saharon Shelah, a proof that Shelah's eventual categoricity conjecture follows from a large cardinal axiom (a proper class of strongly compact cardinals). The main tool is the method of multidimensional diagrams, introduced by Shelah in the eighties and later rediscovered by Zilber in his study of pseudoexponential fields. The main technical result is that (assuming large cardinals), an AEC with unbounded categoricity spectrum is excellent, i.e. any finite diagram of object can be amalgamated in a way that is, in some category-theoretic sense, unique. The categoricity conjecture then follows from excellence. The method seems to have a strongly category-theoretic nature, and has many other applications that do not need large cardinals. For example, assuming that cardinal exponentiation is injective (a weakening of the GCH), one gets a full understanding of the categoricity spectrum of AECs with the amalgamation property. Assuming a little bit more than the GCH, the eventual categoricity conjecture also holds for any AEC with no maximal models.

#### Andres Villaveces

Interpretation and reconstruction in AECs: a blueprint

We provide a blueprint for the study of interpretations of Abstract Elementary Classes: we first revisit interpretability and internality in a category-theoretical language (for first order theories). This reframes work of Hrushovski and Kamensky in a formalism derived from Makkai. We then describe the issue of recovering the biintepretability class of a theory in terms of the automorphism group of a saturated model, and the role of the "Small Index Property" (SIP). An SIP theorem for AECs with strong amalgamation properties we published in 2017 is now placed in the context of reconstruction: we propose a notions of interpretation between some specific kinds of AECs. This is joint work with Zaniar Ghadernezhad.

#### Pedro Zambrano

Tameness in classes of generalized metric structures: quantale-spaces, fuzzy sets, and sheaves

Joint work with Michael Lieberman and Jiri Rosicky Tameness is a very important model-theoretic property of abstract classes of structures, under the assumption of which strong categoricity and stability transfer theorems tend to hold. We generalize the argument of Lieberman and Rosicky---based on Makkai and Paré's result on the accessibility of powerful images of accessible functors under a large cardinal assumption---that tameness holds in classes of metric structures, noting that the argument works just as well for structures with underlying Q-spaces, Q a reasonable quantale. Dropping the reflexivity assumption from the definition of metrics, we obtain a similar result for classes with underlying partial metric spaces: through straightforward translations from partial metrics to fuzzy sets and sheaves, we obtain, respectively, fuzzy and sheafy analogues of this result.