2023/01/16 by Kengo Hirata, Hirata, Kengo
Mathematics · #18N10 #18N15 #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2301.06420
openalex publication_date 2023/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study lax functors between bicategories as a generalized concept of monads and describe generalized notions and theorems of formal monad theory for lax functors. Our first approach is to use the 2-monad whose lax algebras are lax functors. We define lax doctrinal adjunctions for a 2-monad T on a 2-category K, and we show that if K admits and T preserves certain codescent objects, the 2-category Lax-T-Algc of lax algebras and colax morphisms can coreflectively be embedded in the 2-category of lax doctrinal adjunctions. This coreflective embedding generalizes the relation between monads and adjunctions. Our second approach is to see a distributive law for monads as a 2-functor from a lax Gray tensor product, and we show a generalized form of Beck's characterization of distributive laws.