[Paper Review] Augmented virtual double categories

Seerp Roald Koudenburg|arXiv (Cornell University)|Oct 24, 2019

Homotopy and Cohomology in Algebraic Topology4 citations

TL;DR

This paper introduces augmented virtual double categories, extending virtual double categories by allowing cells with empty horizontal targets. It shows that such structures naturally support a notion of 'locally small object' via horizontal units and that unital augmented virtual double categories are equivalent to unital virtual double categories, enabling streamlined proofs and broader applicability in formal category theory.

ABSTRACT

In this article the notion of virtual double category (also known as fc-multicategory) is extended as follows. While cells in a virtual double category classically have a horizontal multi-source and single horizontal target, the notion of augmented virtual double category introduced here extends the latter notion by including cells with empty horizontal target as well. Any augmented virtual double category comes with a built-in notion of "locally small object" and we describe advantages of using augmented virtual double categories as a setting for formal category rather than 2-categories, which are classically equipped with a notion of "admissible object" by means of a yoneda structure in the sense of Street and Walters. An object is locally small precisely if it admits a horizontal unit, and we show that the notions of augmented virtual double category and virtual double category coincide in the presence of all horizontal units. Without assuming the existence of horizontal units we show that most of the basic theory for virtual double categories, such as that of restriction and composition of horizontal morphisms, extends to augmented virtual double categories. We introduce and study in augmented virtual double categories the notion of "pointwise" composition of horizontal morphisms, which formalises the classical composition of profunctors given by the coend formula.

Motivation & Objective

• To extend the theory of virtual double categories by allowing cells with empty horizontal targets, introducing the new structure of augmented virtual double categories.
• To establish that locally small objects in an augmented virtual double category correspond precisely to those admitting a horizontal unit.
• To demonstrate that most foundational theory of virtual double categories—such as restriction and composition of horizontal morphisms—extends to the augmented setting.
• To formalize 'pointwise' composition of horizontal morphisms using coend-like formulas, generalizing profunctor composition.
• To prove that unital augmented virtual double categories are categorically equivalent to unital virtual double categories, unifying two existing frameworks.

Proposed method

• Define augmented virtual double categories as a generalization of virtual double categories that include cells with empty horizontal targets.
• Introduce the notion of 'locally small object' as an object admitting a horizontal unit cell, linking it to representability and Yoneda structures.
• Extend the theory of restriction and composition of horizontal morphisms to the augmented setting, preserving key properties.
• Formalize 'pointwise' horizontal composition via universal cells satisfying a coend-like universal property.
• Construct a strict 2-functor N: VirtDblCatu → AugVirtDblCatu that maps unital virtual double categories to their augmented counterparts.
• Prove that the composition of N with the forgetful functor U: AugVirtDblCat → VirtDblCat is naturally isomorphic to the identity, establishing a 2-categorical equivalence.

Experimental results

Research questions

• RQ1 How can the notion of virtual double category be extended to include cells with empty horizontal targets?
• RQ2 What is the categorical significance of objects admitting a horizontal unit in the context of augmented virtual double categories?
• RQ3 To what extent does the foundational theory of virtual double categories—such as restriction and composition—extend to augmented virtual double categories?
• RQ4 How does pointwise composition of horizontal morphisms in augmented virtual double categories formalize classical profunctor composition via coends?
• RQ5 What is the relationship between unital augmented virtual double categories and unital virtual double categories, and are they categorically equivalent?

Key findings

• The notion of augmented virtual double category extends virtual double categories by including cells with empty horizontal targets, enabling a more natural treatment of vertical morphisms as forming a 2-category.
• An object is locally small if and only if it admits a horizontal unit, establishing a direct categorical characterization of local smallness.
• The basic theory of restriction and composition of horizontal morphisms extends to augmented virtual double categories even without assuming horizontal units.
• The notion of 'pointwise' composition of horizontal morphisms is formalized via universal cells, generalizing the classical coend formula for profunctor composition.
• Unital augmented virtual double categories are categorically equivalent to unital virtual double categories, with a strict 2-functor N: VirtDblCatu → AugVirtDblCatu providing the equivalence.
• The equivalence is realized via a strict 2-functor N and a 2-natural isomorphism τ: id ≅ N∘U, showing that the augmented framework subsumes and simplifies the classical unital case.