Lax algebras via initial monad morphisms: , , and

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 P ¯ 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 P ¯ 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 .