Topics in categorical algebra and Galois theory - Fourie, Jason (Department of Mathematics and Applied Mathematics, 2018)
We provide an overview of the construction of categorical semidirect products and discuss their form in particular semi-abelian varieties. We then give a thorough description of categorical Galois theory, which yields an analogue for the fundamental theorem of Galois theory in an abstract category by making use of the notions of admissibility and effective descent. We show that the admissibility of a functor can be extended to the admissibility of the canonically induced functor on the associated category of pointed objects, and that an analogous extension can be made for effective descent morphisms. We use the extended notions of admissibility and effective descent to describe a new, pointed version of the categorical Galois theorem, and use this result to move the categorical formulation of the Galois theory of finite, separable field extensions into a non-unital context.