Related papers: Notes on limits of accessible categories

In this paper we characterize those accessible $\mathcal V$-categories that have limits of a specified class. We do this by introducing the notion of companion $\mathfrak C$ for a class of weights $\Psi$, as a collection of special types of…

Compact categories have lately seen renewed interest via applications to quantum physics. Being essentially finite-dimensional, they cannot accomodate (co)limit-based constructions. For example, they cannot capture protocols such as quantum…

We provide a complete classification of all the possible categoricity spectra, in terms of internal size, that can appear in a large accessible category with directed colimits, assuming the Singular Cardinal Hypothesis ($SCH$), and…

The paper gives several sufficient conditions on the paracompactness of box products with an arbitrary number of many factors and boxes of arbitrary size. The former include results on generalised metrisability and Sikorski spaces. Of…

We show that metric abstract elementary classes (mAECs) are, in the sense of [LR] (i.e. arXiv:1404.2528), coherent accessible categories with directed colimits, with concrete $\aleph_1$-directed colimits and concrete monomorphisms. More…

For infinite cardinals $\kappa,\lambda$ let $C(\kappa,\lambda)$ denote the class of all compact Hausdorff spaces of weight $\kappa$ and size $\lambda$. So $C(\kappa,\lambda)=\emptyset$ if $\kappa>\lambda$ or $\lambda>2^\kappa$. If F is a…

We show that generalized eventually narrow sequences on a strongly inaccessible cardinal $\kappa$ are preserved under the Cummings-Shaleh non-linear iterations of the higher Hechler forcing on $\kappa$. Moreover assuming GCH,…

In the original version of this paper, we assume a theory $T$ that the logic $\mathbb L_{\kappa, \aleph_{0}}$ is categorical in a cardinal $\lambda > \kappa$, and $\kappa$ is a measurable cardinal. There we prove that the class of model of…

The following pcf results are proved: 1. Assume that kappa > aleph_0 is a weakly compact cardinal. Let mu > 2^kappa be a singular cardinal of cofinality kappa. Then for every regular lambda < pp^+_{Gamma(kappa)} (mu) there is an increasing…

The structure of automorphism groups of $\kappa$-existentially closed groups are studied by Kaya-Kuzucuo\u{g}lu in 2022. It was proved that Aut(G) is the union of subgroups of level preserving automorphisms and $|Aut(G)|=2^\kappa$ whenever…

We give a purely category-theoretic proof of the result of Makkai and Par\'e saying that the category $\bf Lin$ of linearly ordered sets and order preserving injective mappings is a minimal finitely accessible category. We also discuss the…

We prove a correspondence between $\kappa$-small fibrations in simplicial presheaf categories equipped with the injective or projective model structure (and left Bousfield localizations thereof) and relatively $\kappa$-compact maps in their…

Assuming $\kappa$ is a supercompact cardinal and $\lambda$ is an inaccessible cardinal above it, we present an idea due to Magidor, to find a generic extension in which $\kappa=\aleph_\omega$ and $\lambda=\aleph_{\omega+1}.$

Categorical structures and their pseudomaps rarely form locally presentable 2-categories in the sense of Cat-enriched category theory. However, we show that if the categorical structure in question is sufficiently weak (such as the…

Given a cardinal $\kappa$ that is $\lambda$-supercompact for some regular cardinal $\lambda\geq\kappa$ and assuming $\GCH$, we show that one can force the continuum function to agree with any function $F:[\kappa,\lambda]\cap\REG\to\CARD$…

We introduce the notion of $\lambda$-equivalence and $\lambda$-embeddings of objects in suitable categories. This notion specializes to $L_{\infty\lambda}$-equivalence and $L_{\infty\lambda}$-elementary embedding for categories of…

Answering some of the main questions from [MR13], we show that whenever $\kappa$ is a cardinal satisfying $\kappa^{< \kappa} = \kappa > \omega$, then the embeddability relation between $\kappa$-sized structures is strongly invariantly…

Let $\kappa$ be a regular cardinal. We study Gabriel-Ulmer duality when one restricts the 2-category of locally $\kappa$-presentable categories with $\kappa$-accessible right adjoints to its locally full sub-2-category of…

In this note we answer the following question of Grinblat: Is it consistent that for some set X, cov(NULL restriction X)=lambda is a weakly inaccessible cardinal (so X not null of course) while cov(meagre) is small, say it is aleph_1.

We show that if \kappa\ is a weakly compact cardinal then the embeddability relation on (generalized) trees of size \kappa\ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space 2^\kappa\…