Being Van Kampen is a universal property

TL;DR

This paper characterizes Van Kampen cocones as bicolimits in the bicategory of spans, revealing their universal property and exactness features in categories with pullbacks and colimits.

Contribution

It provides a new universal property characterization of Van Kampen cocones using bicategories of spans, connecting colimit conditions with bicategorical bicolimits.

Findings

Van Kampen cocones are characterized as bicolimits in bicategories of spans.

The main theorem links Van Kampen conditions to universal properties in bicategories.

Categories with pullbacks and enough colimits support this bicategorical characterization.

Abstract

Colimits that satisfy the Van Kampen condition have interesting exactness properties. We show that the elementary presentation of the Van Kampen condition is actually a characterisation of a universal property in the associated bicategory of spans. The main theorem states that Van Kampen cocones are precisely those diagrams in a category that induce bicolimit diagrams in its associated bicategory of spans, provided that the category has pullbacks and enough colimits.