Multiple categories of general quintets

TL;DR

This work develops a non-cubical, highly structured framework for higher quintets by systematically constructing multiple categories of generalised quintets, chiral multiple categories, arrow bundles, and chains of adjunctions. Central to the approach is the introduction of mixed laxity morphisms and interchanger coherence, enabling non-uniform directions of arrows and a coskeletal dimension bounded by 3. The paper then builds a non-cubical multiple category of chiral categories, shows how chains of adjunctions and arrow bundles fit into this picture, and finally ties these ideas to pseudo algebras for a 2-monad, producing a rich triple-category of algebras, morphisms, and higher cells. The resulting framework unifies and extends prior double/triple/intercategory theories, providing tools to model complex higher-dimensional categorical structures with both strict and weak interactions. The constructions have implications for understanding generalised adjunctions, intercategories, and algebraic structures in higher category theory.

Abstract

We construct various multiple categories, based on generalised Ehresmann quintets. The main construction is a multiple category whose objects are all the `lax' multiple categories; the transversal arrows are their strict multiple functors while the arrows in a positive direction are multiple functors of a `mixed laxity', varying from the lax ones (in direction 1) to the colax ones (in direction \infty).

Theorems & Definitions (6)

• Theorem 1
• proof
• Theorem 2: Strict double categories as algebras
• proof
• Theorem 3: Weak double categories as normal pseudo algebras
• proof