Free bifibrations of $(\infty,2)$-categories, 2-simplicial objects and the walking adjunction

Fernando Abellán

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arXiv · 2609.17282

Free bifibrations of $(\infty,2)$-categories, 2-simplicial objects and the walking adjunction

Fernando Abellán

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Abstract

In this work, we develop a fibrational approach to freely adjoining adjoints in an $(\infty,2)$-category. We construct the universal bifibration obtained from a cocartesian fibration of $(\infty,2)$-categories by adjoining cartesian lifts over a chosen class of $1$-morphisms in the base. We then use this construction to provide an explicit model for freely adjoining adjoints, together with a zig-zag formula for the resulting mapping $(\infty,1)$-categories. As applications, we give a model-independent proof of the universal property of the walking adjunction, providing an alternative proof of a theorem of Riehl--Verity, and establish the universal characterization of the simplex $2$-category conjectured by Dyckerhoff--Kapranov--Schechtman--Soibelman.

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Fernando Abellán. 2026-09-15. Free bifibrations of $(\infty,2)$-categories, 2-simplicial objects and the walking adjunction. https://arxiv.org/abs/2609.17282

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