Two-variable fibrations, factorisation systems and -categories of spans
Peer reviewed, Journal article
Published version
Date
2023Metadata
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- Institutt for matematiske fag [2530]
- Publikasjoner fra CRIStin - NTNU [38672]
Abstract
We prove a universal property for ∞-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 (∞,2)-category (building on work of Abellán García and Stern), formally identify the unstraightenings of the identity functor on the ∞-category of ∞-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).