This analysis reveals how Barwick's unfurling construction universally adds right adjoints in infinity-categories, indicating an explicit connection to cartesian normed structures.
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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Cnossen et al. (2025) studied this question.
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