Geodesic
Two-variable fibrations, factorisation systems and $\infty $-categories of spans
Forum of Mathematics, Sigma, Tome 11 (2023)

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We prove a universal property for $\infty $-categories of spans in the generality of Barwick’s adequate triples, explicitly describe the cocartesian fibration corresponding to the span functor, and show that the latter restricts to a self-equivalence on the class of orthogonal adequate triples, which we introduce for this purpose. As applications of the machinery we develop, we give a quick proof of Barwick’s unfurling theorem, show that an orthogonal factorisation system arises from a cartesian fibration if and only if it forms an adequate triple (generalising work of Lanari), extend the description of dual (co)cartesian fibrations by Barwick, Glasman and Nardin to two-variable fibrations, explicitly describe parametrised adjoints (extending work of Torii), identify the orthofibration classifying the mapping category functor of an $(\infty ,2)$-category (building on work of Abellán García and Stern), formally identify the unstraightenings of the identity functor on the $\infty $-category of $\infty $-categories with the (op)lax under-categories of a point, and deduce a certain naturality property of the Yoneda embedding (answering a question of Clausen). 
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     author = {Rune Haugseng and Fabian Hebestreit and Sil Linskens and Joost Nuiten},
     title = {Two-variable fibrations, factorisation systems and $\infty $-categories of spans},
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Rune Haugseng; Fabian Hebestreit; Sil Linskens; Joost Nuiten. Two-variable fibrations, factorisation systems and $\infty $-categories of spans. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.107

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