Higher Groups and Higher Normality

TL;DR

This paper advances homotopical group theory in $mbda$-topoi by generalizing normal maps, monads, and classical theorems, including a new Orbit-Stabilizer result for $mbda$-topoi.

Contribution

It introduces a generalized framework for homotopy normal maps, extends monoidal functor properties, and formulates Noether's Theorems in the context of $mbda$-topoi.

Findings

Normal maps are algebras for a 'normal closure' monad.

Monoidal functors preserve normal maps and fiber properties.

Established an Orbit-Stabilizer Theorem in $mbda$-topoi.

Abstract

In this paper we continue Prasma's homotopical group theory program by considering homotopy normal maps in arbitrary $\infty$-topoi. We show that maps of group objects equipped with normality data, in Prasma's sense, are algebras for a "normal closure" monad in a way which generalizes the standard loops-suspension monad. We generalize a result of Prasma by showing that monoidal functors of $\infty$-topoi preserve normal maps or, equivalently, that monoidal functors of $\infty$-topoi preserve the property of "being a fiber" for morphisms between connected objects. We also formulate Noether's Isomorphism Theorems in this setting, prove the first of them, and provide counterexamples to the other two. Accomplishing these goals requires us to spend substantial time synthesizing existing work of Lurie so that we may rigorously talk about group objects in $\infty$-topoi in the "usual way." One nice result of this labor is the formulation and proof of an Orbit-Stabilizer Theorem for group actions in $\infty$-topoi.