Lax algebras via initial monad morphisms: APP, TOP, MET and ORD

This paper contributes to the algebraization of topology via the theory of monads and lax extensions of monads and their associated lax algebras (see Barr (1970) [1], , Clementino and Hofmann (2003) [2], , Clementino, Hofmann and Tholen (2004) [4], , Clementino and Tholen (2003) [5], , Lowen and Vroegrijk (2008) [11], , Manes (1974) [12], , Seal (2005) [14], ). We construct a monad P, a lax extension and monad morphisms into P from the most important monads as studied in the aforementioned papers such that their lax extensions and their associated categories of lax algebras can be derived from the extension by initial lifts via these monad morphisms. This provides us with a completely unified way to obtain the categories Top, App, Met and Ord without the necessity to leave the realm of Rel as was previously required in Clementino and Hofmann (2003) [2], , Clementino, Hofmann and Tholen (2004) [4], and Clementino and Tholen (2003) [5] in particular in order to obtain App and Met.