2024/12/10 by Nathanael Arkor, Arkor, Nathanael, John Bourke +3 · 2 voices · 2 citations
Computer Science · Mathematics · #18C10 #18C30 #18C40 #18D20 #18M65 #18N10 #Category Theory (math.CT) #Data Visualization and Analytics #FOS: Mathematics #math.CT
paper · pdf · doi:10.48550/arxiv.2412.07475
openalex publication_date 2024/12/10 · arxiv published 2024/12/10 · arxiv updated 2024/12/10 · openalex created_date 2024/12/12 · openalex updated_date 2026/08/01
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.