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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.18278 |
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| _version_ | 1866911160076337152 |
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| author | Cnossen, Bastiaan Lenz, Tobias Ramzi, Maxime |
| author_facet | Cnossen, Bastiaan Lenz, Tobias Ramzi, Maxime |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18278 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Universality of Barwick's unfurling construction Cnossen, Bastiaan Lenz, Tobias Ramzi, Maxime Algebraic Topology Category Theory 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. |
| title | Universality of Barwick's unfurling construction |
| topic | Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2502.18278 |