Admissibility, stable units and connected components

Abstract

Consider a reflection from a finitely-complete category $\mathbb{C}$ into its full subcategory $\mathbb{M}$, with unit $η:1_\mathbb{C}\rightarrow HI$. Suppose there is a left-exact functor $U$ into the category of sets, such that $UH$ reflects isomorphisms and $U(η_C)$ is a surjection, for every $C\in\mathbb{C}$. If, in addition, all the maps $\mathbb{M}(T,M)\rightarrow \mathbf{Set}(1,U(M))$ induced by the functor $UH$ are surjections, where $T$ and 1 are respectively terminal objects in $\mathbb{C}$ and $\mathbf{Set}$, for every object $M$ in the full subcategory $\mathbb{M}$, then it is true that: the reflection $H\vdash I$ is semi-left-exact (admissible in the sense of categorical Galois theory) if and only if its connected components are "connected"; it has stable units if and only if any finite product of connected components is "connected". Where the meaning of "connected" is the usual in categorical Galois theory, and the definition of connected component with respect to the ground structure will be given. Note that both algebraic and topological instances of Galois structures are unified in this common setting, with respect to categorical Galois theory.