2016/03/20 by Walter Tholen, Tholen, Walter
Mathematics · Medicine · #18 C15 #18C20 #18D99 #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Intracranial Aneurysms: Treatment and Complications #math.CT #msc:18 #msc:18C20 #msc:18D99 #msc:C15
paper · pdf · doi:10.48550/arxiv.1603.06251
32 pages
openalex publication_date 2016/03/20 · arxiv created 2016/08/23 · arxiv updated 2016/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a quantaloid Q, considered as a bicategory, Walters introduced categories enriched in Q. Here we extend the study of monad-quantale-enriched categories of the past fifteen years by introducing monad-quantaloid-enriched categories. We do so by making lax distributive laws of a monad \mathbbT over the discrete presheaf monad of the small quantaloid Q the primary data of the theory, rather than the lax monad extensions of \mathbbT to the category of Q-relations that they equivalently describe. The central piece of the paper establishes a Galois correspondence between such lax distributive laws and lax Eilenberg-Moore \mathbbT-algebra structures on the set of discrete presheaves over the object set of Q. We give a precise comparison of these structures with the more restrictive notion introduced by Hofmann in the case of a commutative quantale, called natural topological theories here, and describe the lax monad extensions introduced by him as minimal. Throughout the paper, a variety of old and new examples of ordered, metric and topological structures illustrate the theory developed, which includes the consideration of algebraic functors and change-of-base functors in full generality.