Related papers: $\infty$-topoi and Natural Phenomena: Generation
Self-regulatory models are common in nature, as described e.g. in (\cite{mur}), (\cite{ha}) and (\cite{Gb}).\\ Let us consider a system made up of a number of glands as a motivation. Each gland secretes a hormone that allows secretion in…
Recent advancements in graph representation learning have shifted attention towards dynamic graphs, which exhibit evolving topologies and features over time. The increased use of such graphs creates a paramount need for generative models…
In this survey, we provide an overview of category theory-derived machine learning from four mainstream perspectives: gradient-based learning, probability-based learning, invariance and equivalence-based learning, and topos-based learning.…
The classification of one parameter local Coulomb branch solution of theories with eight supercharges is given by assuming that it is given by a genus $g$ fiberation of Riemann surfaces. The crucial point is the fact that certain conjugacy…
We classify thick subcategories of the $\infty$-categories of perfect modules over ring spectra which arise as functions on even periodic derived stacks satisfying affineness and regularity conditions. For example, we show that the thick…
A locally connected topos is a Galois topos if the Galois objects generate the topos. We show that the full subcategory of Galois objects in any connected locally connected topos is an inversely 2-filtered 2-category, and as an application…
Recently, we have endowed various categories of groups with topologies. The purpose of this paper is to introduce on these categories others topologies which are statistically more suitable to study well-known problems in groups theory. We…
We use double categories to obtain a single theorem characterizing certain exponentiable morphisms of small categories, topological spaces, locales, and posets.
Let {\phi} be an automorphism on a connected Lie group G. Through several G-subgroups associated to the dynamics of {\phi} we analyze their topological entropy. Assume that G belongs to the class of finite semisimple center Lie groups which…
Given a good homology theory E and a topological space X, the E-homology of X is not just an E_{*}-module but also a comodule over the Hopf algebroid (E_{*}, E_{*}E). We establish a framework for studying the homological algebra of…
We wish to investigate some elementary problems concerning topological dynamics revolving around our proposed definition of escaping set. We also discuss the notion of escaping set in the induced dynamics of the hyperspace. Moreover, we…
We study the relationship between partial map classifiers, Sierpi\'nski cones, and axioms for synthetic higher categories and domains within univalent foundations. In particular, we show that synthetic $\infty$-categories are closed under…
Since the time when the first optical instruments have been invented, an idea that the visible image of an object under observation depends on tools of observation became commonly assumed in physics. A way to formalize it in mathematics is…
We propose a means to relate properties of an interconnected system to its separate component systems in the presence of cascade-like phenomena. Building on a theory of interconnection reminiscent of the behavioral approach to systems…
Text-to-image generative models have enabled high-resolution image synthesis across different domains, but require users to specify the content they wish to generate. In this paper, we consider the inverse problem -- given a collection of…
We find a strong separation between two natural families of simple rank one theories in Keisler's order: the theories $T_\mathfrak{m}$ reflecting graph sequences, which witness that Keisler's order has the maximum number of classes, and the…
We further develop Weber's notion of elementary 2-topos by proposing certain new axioms. We show that in a 2-category C satisfying these axioms, the "discrete opfibration (DOF) classifier" S is always an internal elementary 1-topos, in an…
We apply, in the context of semigroups, the main theorem from~\cite{higjac} that an elementary class $\mathcal{C}$ of algebras which is closed under the taking of direct products and homomorphic images is defined by systems of equations. We…
We compute rationally the topological (complex) K-theory of the classifying space BG of a discrete group provided that G has a cocompact G-CW-model for its classifying space for proper G-actions. For instance word-hyperbolic groups and…
We classify all subgroups of $SO(3)$ that are generated by two elements, each a rotation of finite order, about axes separated by an angle that is a rational multiple of $\pi$. In all cases we give a presentation of the subgroup. In most…