Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation

TL;DR

This paper explores the role of strictness in two-dimensional category theory structures, introducing enhanced 2-categories and limit sketches to explain their properties and symmetry of internalisation.

Contribution

It introduces enhanced 2-categorical limit sketches and demonstrates the symmetry of internalisation in two-dimensional categorical structures.

Findings

Enhanced 2-categories clarify strictness phenomena.

Established symmetry of internalisation for such structures.

Revealed equivalences between complex categorical structures.

Abstract

Many structures of interest in two-dimensional category theory have aspects that are inherently strict. This strictness is not a limitation, but rather plays a fundamental role in the theory of such structures. For instance, a monoidal fibration is - crucially - a strict monoidal functor, rather than a pseudo or lax monoidal functor. Other examples include monoidal double categories, double fibrations, and intercategories. We provide an explanation for this phenomenon from the perspective of enhanced 2-categories, which are 2-categories having a distinguished subclass of 1-cells representing the strict morphisms. As part of our development, we introduce enhanced 2-categorical limit sketches and explain how this setting addresses shortcomings in the theory of 2-categorical limit sketches. In particular, we establish the symmetry of internalisation for such structures, entailing, for instance, that a monoidal double category is equivalently a pseudomonoid in an enhanced 2-category of double categories, or a pseudocategory in an enhanced 2-category of monoidal categories.