Universality of span 2-categories and the construction of 6-functor formalisms

TL;DR

This paper establishes a universal property of higher span 2-categories, linking biadjointable functors with 6-functor formalisms, and extends the framework to symmetric monoidal contexts.

Contribution

It provides a conceptual and independent proof of the Mann-Liu-Zheng construction of 6-functor formalisms using higher span 2-categories.

Findings

Higher span 2-categories have a universal property relating to biadjointable functors.

The universality extends to symmetric monoidal and lax symmetric monoidal settings.

The framework offers a new perspective on constructing 6-functor formalisms.

Abstract

Given an $\infty$-category $C$ equipped with suitable wide subcategories $I, P \subset E\subset C$, we show that the $(\infty,2)$-category $\text{S}{\scriptstyle\text{PAN}}_2(C,E)_{P,I}$ of higher (or iterated) spans defined by Haugseng has the universal property that 2-functors $\text{S}{\scriptstyle\text{PAN}}_2(C,E)_{P,I} \to \mathbb D$ correspond precisely to $(I, P)$-biadjointable functors $C^\text{op} \to \mathbb D$, i.e. functors $F$ where $F(i)$ for $i \in I$ admits a left adjoint and $F(p)$ for $p \in P$ admits a right adjoint satisfying various Beck-Chevalley conditions. We also extend this universality to the symmetric monoidal and lax symmetric monoidal settings. This provides a conceptual explanation for - and an independent proof of - the Mann-Liu-Zheng construction of 6-functor formalisms from suitable functors $C^\text{op}\to\text{CAlg}(\text{Cat})$.