Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFraisse sequences: category-theoretic approach to universal homogeneous structures
271
0
0.0
(
0
)
Authors
Wieslaw Kubis
Ask ChatGPT about the research
No Arabic abstract
We develop category-theoretic framework for universal homogeneous objects, with some applications in the theory of Banach spaces, linear orderings, and in topology of compact spaces.
rate research
Read More
We study topological groups $G$ for which the universal minimal $G$-system $M(G)$, or the universal irreducible affine $G$-system $IA(G)$ are tame. We call such groups intrinsically tame and convexly intrinsically tame. These notions are generaliz
Quasi-abelian categories are abundant in functional analysis and representation theory. It is known that a quasi-abelian category $mathcal{E}$ is a cotilting torsionfree class of an abelian category. In fact, this property characterizes quasi-abelian categories. This ambient abelian category is derived equivalent to the category $mathcal{E}$, and can be constructed as the heart $mathcal{LH}(mathcal{E})$ of a $operatorname{t}$-structure on the bounded derived category $operatorname{D^b}(mathcal{E})$ or as the localization of the category of monomorphisms in $mathcal{E}.$ However, there are natural examples of categories in functional analysis which are not quasi-abelian, but merely one-sided quasi-abelian or even weaker. Examples are the category of $operatorname{LB}$-spaces or the category of complete Hausdorff locally convex spaces. In this paper, we consider additive regular categories as a generalization of quasi-abelian categories that covers the aforementioned examples. These categories can be characterized as pre-torsionfree subcategories of abelian categories. As for quasi-abelian categories, we show that such an ambient abelian category of an additive regular category $mathcal{E}$ can be found as the heart of a $operatorname{t}$-structure on the bounded derived category $operatorname{D^b}(mathcal{E})$, or as the localization of the category of monomorphisms of $mathcal{E}$. In our proof of this last construction, we formulate and prove a version of Auslanders formula for additive regular categories. Whereas a quasi-abelian category is an exact category in a natural way, an additive regular category has a natural one-sided exact structure. Such a one-sided exact category can be 2-universally embedded into its exact hull. We show that the exact hull of an additive regular category is again an additive regular category.
This paper addresses the growing need to process non-Euclidean data, by introducing a geometric deep learning (GDL) framework for building universal feedforward-type models compatible with differentiable manifold geometries. We show that our GDL models can approximate any continuous target function uniformly on compacts of a controlled maximum diameter. We obtain curvature dependant lower-bounds on this maximum diameter and upper-bounds on the depth of our approximating GDL models. Conversely, we find that there is always a continuous function between any two non-degenerate compact manifolds that any locally-defined GDL model cannot uniformly approximate. Our last main result identifies data-dependent conditions guaranteeing that the GDL model implementing our approximation breaks the curse of dimensionality. We find that any real-world (i.e. finite) dataset always satisfies our condition and, conversely, any dataset satisfies our requirement if the target function is smooth. As applications, we confirm the universal approximation capabilities of the following GDL models: Ganea et al. (2018)s hyperbolic feedforward networks, the architecture implementing Krishnan et al. (2015)s deep Kalman-Filter, and deep softmax classifiers. We build universal extensions/variants of: the SPD-matrix regressor of Meyer et al. (2011), and Fletcher et al. (2009)s Procrustean regressor. In the Euclidean setting, our results imply a quantitative version of Kidger and Lyons (2020)s approximation theorem and a data-dependent version of Yarotsky and Zhevnerchuk (2020)s uncursed approximation rates.
Let $U_q(mathfrak{g})$ be a quantum affine algebra of untwisted affine ADE type and let $mathcal{C}^0_{mathfrak{g}}$ be Hernandez-Leclercs category. For a duality datum $mathcal{D}$ in $mathcal{C}^0_{mathfrak{g}}$, we denote by $mathcal{F}_{mathcal{D}}$ the quantum affine Weyl-Schur duality functor. We give sufficient conditions for a duality datum $mathcal{D}$ to provide the functor $mathcal{F}_{mathcal{D}}$ sending simple modules to simple modules. Then we introduce the notion of cuspidal modules in $mathcal{C}^0_{mathfrak{g}}$, and show that all simple modules in $mathcal{C}^0_{mathfrak{g}}$ can be constructed as the heads of ordered tensor products of cuspidal modules.
In The factorization of the Giry monad (arXiv:1707.00488v2) the author asserts that the category of convex spaces is equivalent to the category of Eilenberg-Moore algebras over the Giry monad. Some of the statements employed in the proof of this claim have been refuted in our earlier paper (arXiv:1803.07956). Building on the results of that paper we prove that no such equivalence exists and a parallel statement is proved for the category of super convex spaces.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
