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Publication# Hopf algebras and Hopf-Galois extensions in infinity-categories

Abstract

In this thesis, we study interactions between algebraic and coalgebraic structures in infinity-categories (more precisely, in the quasicategorical model of (infinity, 1)-categories). We define a notion of a Hopf algebra H in an E-2-monoidal infinity-category and lift some results about ordinary Hopf algebras, such as the fundamental theorem of Hopf modules, to this setting. We also study Hopf-Galois extensions in this context. Given a candidate Hopf-Galois extension, i.e., a map f : A -> B of H-comodule algebras where H coacts on A trivially, we construct a structured version of the comparison map B (x)_A B -> H (x) B that allows us to compare the category of descent data for f with a category of "B-modules equipped with a semilinear coaction of H". We provide further insights into the case of commutative (i.e., E-infinity) comodule algebras over a commutative Hopf algebra, for instance a description of the aforementioned category of modules equipped with a semilinear coaction as the limit of a "categorified cobar construction". Moreover, we provide a simple description of comodules over a space in slice categories of the infinity-category of spaces, which enables us to realize multiplicative Thom objects as comodule algebras and thus incorporate them into the aforementioned framework.

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Related publications (1)

Related concepts (17)

Quasi-category

In mathematics, more specifically , a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a . The study of such generalizations is known as . Quasi-categories were introduced by . André Joyal has much advanced the study of quasi-categories showing that most of the usual basic and some of the advanced notions and theorems have their analogues for quasi-categories.

Infinity

Infinity is something which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes.

Hopf algebra

In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations.

K-Theory was originally defined by Grothendieck as a contravariant functor from a subcategory of schemes to abelian groups, known today as K0. The same kind of construction was then applied to other f