Two-variable fibrations, factorisation systems and $\infty$-categories of spans

TL;DR

This paper develops a universal property for $ abla$-categories of spans within $ abla$-categories, describing their fibrations, dualities, and adjoints, and applies these to various advanced $ abla$-category theories.

Contribution

It introduces a universal property for $ abla$-categories of spans, describes related fibrations, and extends several existing theorems in $ abla$-category theory.

Findings

Provides a quick proof of Barwick's unfurling theorem

Shows orthogonal factorisation systems arise from cartesian fibrations if and only if they form an adequate triple

Identifies the unstraightenings of the identity functor with (op)lax under-categories

Abstract

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 Abell\'an Garcia 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).