Universality of Barwick's unfurling construction
| Title: | Universality of Barwick's unfurling construction |
|---|---|
| Authors: | Cnossen, Bastiaan, Lenz, Tobias, Ramzi, Maxime |
| Publication Year: | 2025 |
| Collection: | Mathematics |
| Subject Terms: | Mathematics - Algebraic Topology, Mathematics - Category Theory |
| More Details: | Given an $\infty$-category $\mathcal{C}$ with pullbacks, its $(\infty,2)$-category $\mathbf{Span}(\mathcal{C})$ of spans has the universal property of freely adding right adjoints to morphisms in $\mathcal{C}$ satisfying a Beck--Chevalley condition. We show that this universal property is implemented by an $(\infty,2)$-categorical refinement of Barwick's \emph{unfurling construction}: For any right adjointable functor $\mathcal{C} \to \mathrm{Cat}_{\infty}$, the unstraightening of its unique extension to $\mathbf{Span}(\mathcal{C})$ can be explicitly written down as another span $(\infty,2)$-category, and on underlying $(\infty,1)$-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. Comment: 14 pages |
| Document Type: | Working Paper |
| Access URL: | http://arxiv.org/abs/2502.18278 |
| Accession Number: | edsarx.2502.18278 |
| Database: | arXiv |
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