2023/11/27 by Adrian Miranda, Miranda, Adrian
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2311.15618
openalex publication_date 2023/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A pseudomonad on a 2-category whose underlying endomorphism is a 2-functor can be seen as a diagram Psmnd → Gray for which weighted limits and colimits can be considered. The 2-category of pseudoalgebras, pseudomorphisms and 2-cells is such a Gray-enriched weighted limit \citeCoherent Approach to Pseudomonads, however neither the Kleisli bicategory nor the 2-category of free pseudoalgebras are the analogous weighted colimit \citeFormal Theory of Pseudomonads. In this paper we describe the actual weighted colimit via a presentation, and show that the comparison 2-functor induced by any other pseudoadjunction splitting the original pseudomonad is bi-fully faithful. As a consequence, we see that biessential surjectivity on objects characterises left pseudoadjoints whose codomains have an `up to biequivalence' version of the universal property for Kleisli objects. This motivates a homotopical study of Kleisli objects for pseudomonads, and to this end we show that the weight for Kleisli objects is cofibrant in the projective model structure on [Psmndop, Gray].