  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.
