Presheaves and cocompletions in formal category theory

TL;DR

This paper explores the connection between presheaf constructions and free cocompletions in formal category theory, extending weighted colimits and providing new methods for enriched categories.

Contribution

It establishes the equivalence of presheaf constructions and free cocompletions under mild assumptions and extends weighted colimits theory to enriched and bicategory contexts.

Findings

Presheaf constructions coincide with free cocompletions in certain settings.

Developed recognition theorems for presheaf objects and cocompletions.

Constructed free cocompletions for enriched categories and bicategories.

Abstract

We study the relationship between presheaf constructions and free cocompletions in the context of formal category theory, elucidating the coincidence between the two concepts in familiar settings. We show that, in a virtual equipment satisfying mild assumptions, free cocompletions under classes of weights are exhibited by presheaf constructions. We furthermore extend the theory of weighted colimits from enriched category theory to this setting, developing the concepts of atomicity and rank, and providing recognition theorems for presheaf objects, free cocompletions, and cocomplete objects. As an application of our methods, we construct free cocompletions, under arbitrary classes of colimit-small weights, of (possibly large) categories enriched in (not necessarily symmetric) monoidal categories and bicategories; this resolves a longstanding omission in the literature on enriched category theory.