On Normal Maps and Noether's Isomorphism Theorems for ∞-Groups
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Authors
Fox, Landon
Issue Date
2024
Type
Thesis
Language
Keywords
Algebraic Topology , Higher Algebra , Higher Category Theory , Infinity-groups , Loop Spaces
Alternative Title
Abstract
In this thesis we further develop the notion of homotopy normal maps of loop spaces introduced by Prasma. This differs from the classical notion of normality in group theory in that the property of being a normal subgroup is replaced with a normality datum. In particular, we investigate the relationship between these ideas and Noether's three isomorphism theorems in classical group theory. We show that an analog of Noether's first isomorphism always holds; the second does not hold in general; and we provide sufficient conditions for which the third isomorphism theorem holds in the context of ∞-groups. In the process of proving these claims we develop a factorization system on the ∞-category of ∞-groups and relate it to the classical epimorphism/monomorphism factorization of ordinary groups.