Publication

A Hopf algebra model for Dwyer's tame spaces

Haoqing Wu
2022
EPFL thesis
Abstract

In this thesis, we give a modern treatment of Dwyer's tame homotopy theory using the language of \infty-categories.We introduce the notion of tame spectra and show it has a concrete algebraic description.We then carry out a study of \infty-operads and define tame spectral Lie algebras and tame spectral Hopf algebras. Finally, we prove that the homotopy theory of tame spectral Hopf algebras is equivalent to that of tame spaces. To recover Dwyer's Lie algebra model for tame spaces, we use Koszul duality to construct a universal enveloping algebra functor, and show it is an equivalence from the \infty-category of tame spectral Lie algebras to the \infty-category of tame spectral Hopf algebras.

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Algebra
Algebra () is the study of variables and the rules for manipulating these variables in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields.
Homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A1 homotopy theory) and (specifically the study of ). In homotopy theory and algebraic topology, the word "space" denotes a topological space.
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