Enhanced 2-categories and limits for lax morphisms

Abstract

We study limits in 2-categories whose objects are categories with extra structure and whose morphisms are functors preserving the structure only up to a coherent comparison map, which may or may not be required to be invertible. This is done using the framework of 2-monads. In order to characterize the limits which exist in this context, we need to consider also the functors which do strictly preserve the extra structure. We show how such a 2-category of weak morphisms which is "enhanced", by specifying which of these weak morphisms are actually strict, can be thought of as category enriched over a particular base cartesian closed category F. We give a complete characterization, in terms of F-enriched category theory, of the limits which exist in such 2-categories of categories with extra structure.

77 pages; v2 minor changes only, to appear in Advances

Cited by in corpus (22)

• Skew-enriched categories
• Constructing symmetric monoidal bicategories functorially
• Exact completions and small sheaves
• Extending homotopy theories across adjunctions
• The formal theory of relative monads
• Accessible aspects of 2-category theory
• Lax orthogonal factorisation systems
• Contravariance through enrichment
• Enriched categories as a free cocompletion
• Algebraic weighted colimits
• Fibrations of predicates and bicategories of relations
• A general limit lifting theorem for 2-dimensional monad theory
• Monads and distributive laws for Rota-Baxter and differential algebras
• Amortized Analysis via Coalgebra
• On semiflexible, flexible and pie algebras
• Two-dimensional regularity and exactness
• Optics for Premonoidal Categories
• Lifting PIE limits with strict projections
• A colimit decomposition for homotopy algebras in Cat
• Not every pseudoalgebra is equivalent to a strict one
• Transposing cartesian and other structure in double categories
• Two dimensional monadicity