ﻻ يوجد ملخص باللغة العربية
Univalence was first defined in the setting of homotopy type theory by Voevodsky, who also (along with Kapulkin and Lumsdaine) adapted it to a model categorical setting, which was subsequently generalized to locally Cartesian closed presentable $infty$-categories by Gepner and Kock. These definitions were used to characterize various $infty$-categories as models of type theories. We give a definition for univalent morphisms in finitely complete $infty$-categories that generalizes the aforementioned definitions and completely focuses on the $infty$-categorical aspects, characterizing it via representability of certain functors, which should remind the reader of concepts such as adjunctions or limits. We then prove that in a locally Cartesian closed $infty$-category (that is not necessarily presentable) univalence of a morphism is equivalent to the completeness of a certain Segal object we construct out of the morphism, characterizing univalence via internal $infty$-categories, which had been considered in a strict setting by Stenzel. We use these results to study the connection between univalence and elementary topos theory. We also study univalent morphisms in the category of groups, the $infty$-category of $infty$-categories, and pointed $infty$-categories.
We study properties of the category of modules of an algebra object A in a tensor category C. We show that the module category inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a by-prod
We generalize toposic Galois theory to higher topoi. We show that locally constant sheaves in a locally (n-1)-connected n-topos are equivalent to representations of its fundamental pro-n-groupoid, and that the latter can be described in terms of Galo
Every ring extension of $A$ by $R$ induces a pair of group homomorphisms $mathcal{L}^{*}:Rto End_Z(A)/L(A);mathcal{R}^{*}:Rto End_Z(A)/R(A),$ preserving multiplication, satisfying some certain conditions. A such 4-tuple $(R,A,mathcal{L}^{*},mathcal{R
These notes were originally developed as lecture notes for a category theory course. They should be well-suited to anyone that wants to learn category theory from scratch and has a scientific mind. There is no need to know advanced mathematics, nor a
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 clai