Related papers: Universal abstract elementary classes and locally …
We define a natural class of graphs by generalizing prior notions of visibility, allowing the representing regions and sightlines to be arbitrary. We consider mainly the case of compact connected representing regions, proving two results…
We generalise the notion of separable equivalence, originally presented by Linckelmann (2011), to an equivalence relation on additive categories. We use this generalisation to show that from an initial equivalence between two algebras we…
In the setup of abstract elementary classes satisfying a local version of superstability, we prove the uniqueness property for $\mu$-forking, a certain independence notion arising from splitting. This had been a longstanding technical…
Algebraic theories, sometimes called equational theories, are syntactic notions given by finitary operations and equations, such as monoids, groups, and rings. There is a well-known category-theoretic treatment of them that algebraic…
We discuss the homological algebra of representation theory of finite dimensional algebras and finite groups. We present various methods for the construction and the study of equivalences of derived categories: local group theory, geometry…
We generalize a well-known theorem binding the elementary equivalence relation on the level of PAC fields and the isomorphism class of their absolute Galois groups. Our results concern two cases: saturated PAC structures and non-saturated…
Order types are a well known abstraction of combinatorial properties of a point set. By Mn\"ev's universality theorem for each semi-algebraic set $V$ there is an order type with a realization space that is \emph{stably equivalent} to $V$.…
We study a topological analogue of id\`elic class field theory for 3-manifolds, in the spirit of arithmetic topology. We firstly introduce the notion of a very admissible link $\mathcal{K}$ in a 3-manifold $M$, which plays a role analogous…
Algebraic theories with dependency between sorts form the structural core of Martin-L\"of type theory and similar systems. Their denotational semantics are typically studied using categorical techniques; many different categorical…
We introduce a notion of a filtered model structure and use this notion to produce various model structures on pro-categories. This framework generalizes several known examples. We give several examples, including a homotopy theory for…
We provide a proof, in $ZFC$, of Shelah's eventual categoricity conjecture for abstract elementary classes (AEC's). Moreover, assuming in addition the Singular Cardinal Hypothesis ($SCH$), we prove a direct generalization to the more…
In this note we characterize, within the framework of the theory of finite set, those categories of graphs that are {\em algebraic universal} in the sense that every concrete category embeds in them. The proof of the characterization is…
For a locally compact group $G$, we show that it is possible to present the class of continuous unitary representations of $G$ as an elementary class of metric structures, in the sense of continuous logic. More precisely, we show how…
We propose a method that learns a discriminative yet semantic space for object categorization, where we also embed auxiliary semantic entities such as supercategories and attributes. Contrary to prior work which only utilized them as side…
These notes give an elementary introduction to Lie groups, Lie algebras, and their representations. Designed to be accessible to graduate students in mathematics or physics, they have a minimum of prerequisites. Topics include definitions…
This short introductory category theory textbook is for readers with relatively little mathematical background (e.g. the first half of an undergraduate mathematics degree). At its heart is the concept of a universal property, important…
A new case of Shelah's eventual categoricity conjecture is established: $\mathbf{Theorem}$ Let $K$ be an AEC with amalgamation. Write $H_2 := \beth_{\left(2^{\beth_{\left(2^{\text{LS} (K)}\right)^+}}\right)^+}$. Assume that $K$ is…
While many different models for $(\infty,1)$-categories are currently being used, it is known that they are Quillen equivalent to one another. Several higher-order analogues of them are being developed as models for $(\infty,…
Representations are essential to mathematically model phenomena, but there are many options available. While each of those options provides useful properties with which to solve problems related to the phenomena in study, comparing results…
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…