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October 16, 2025Open Access

Universality of Barwick's unfurling construction

Key Points

Universality of Barwick's unfurling construction establishes a framework for right adjoint additions in infinity-categories.
The constructed $( extinfty,2)$-category facilitates the implementation of these right adjoints through spans and ensures compatibility with the Beck--Chevalley condition.
An explicit formulation of the unstraightening process highlights the relationship between infinity-categories and traditional $( extinfty,1)$-categories.
Nardin--Shah and other key constructions within cartesian normed structures are shown to coincide, affirming the relevance of this framework.

Abstract

Given an -category C with pullbacks, its (, 2) -category Span (C) of spans has the universal property of freely adding right adjoints to morphisms in C satisfying a Beck--Chevalley condition. We show that this universal property is implemented by an (, 2) -categorical refinement of Barwick's unfurling construction: For any right adjointable functor C Cat_, the unstraightening of its unique extension to Span (C) can be explicitly written down as another span (, 2) -category, and on underlying (, 1) -categories this recovers Barwick's construction. As an application, we show that the constructions of cartesian normed structures by Nardin--Shah and Cnossen--Haugseng--Lenz--Linskens coincide.

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