From lax monad extensions to Topological theories

Abstract

We investigate those lax extensions of a Set-monad T = (T,m, e) to the category V-Rel of sets and V-valued relations for a quantale V = (V,⊗, k) that are fully determined by ξ maps : TV ⟶ V. We pay special attention to those maps ξ that make V a T-algebra and, in fact, (V,⊗, k) a monoid in the category Set^T with its cartesian structure. Any such map ξ forms the main ingredient to Hofmann’s notion of topological theory.