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Free bifibrations of (∞,2)-categories, 2-simplicial objects and the walking adjunction
Abstract
In this work, we develop a fibrational approach to freely adjoining adjoints in an (∞,2)-category. We construct the universal bifibration obtained from a cocartesian fibration of (∞,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 (∞,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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BibTeX
@misc{abellan2026free,
title = {{Free bifibrations of $(\infty,2)$-categories, 2-simplicial objects and the walking adjunction}},
author = {Fernando Abellán},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17282},
eprint = {2609.17282},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17282},
}
RIS
TY - UNPB
TI - Free bifibrations of (∞,2)-categories, 2-simplicial objects and the walking adjunction
AU - Fernando Abellán
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17282
UR - https://arxiv.org/abs/2609.17282
AB - In this work, we develop a fibrational approach to freely adjoining adjoints in an (∞,2)-category. We construct the universal bifibration obtained from a cocartesian fibration of (∞,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 (∞,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.
ER -