Lax Algebras: A Scenic Approach

Lax algebras provide a setting for the simultaneous study of a wide range of topological structures. We propose scenes as an axiomatic setting for the study of lax algebras.After collecting basic facts on order-enriched categories, we present various methods to construct (lax) extensions of functors. We define lax algebras in a scenic context and present various examples. Exactness properties of categories of lax algebras are discussed. Here, special emphasis is put on strict and costrict morphisms, which provide a generalization of proper and open maps of topological spaces. Using strict morphisms a notion of compactness for lax algebras is discussed. For this notion of compactness a general Tychonoff-type theorem is proved.