Hölder categories (Q2877059)

In this article the authors introduce the notion of a Hölder category, a complete and well powered category in which the initial object is simple, and which has a simple quasi-initial coseparator. The authors prove that when the initial object is a simple coseparator, then every uniformly nontrivial reflection is a monoreflection. (By a uniformly nontrivial reflection is meant a reflection in which the reflection of each non-terminal object is non-terminal). This generalizes a similar result of \textit{G. Bezhanishvili}, \textit{P.~J. Morandi} and \textit{B. Olberding} [Theory Appl. Categ. 28, 435--475, (2013; Zbl 1314.06020)] about reflective subcategories of the category of bounded archimedean \(\ell \)-algebras. The authors also establish that if in a Hölder category every epireflection is a monoreflection, then the initial object is a coseparator.