PreprintCC BY 4.0arXiv
The span-squares adjunction
Abstract
We show a universal property of the span ∞-category that yields a description of functors defined on this category. For this, we view the span construction as a functor from double ∞-categories to ∞-categories, and show that this functor admits a right adjoint defined by the double ∞-categories of squares. As an application, we obtain a model for the localization of an ∞-category in terms of a span ∞-category and also give another proof of the model given by the relative nerve. Moreover, using the span-squares adjunction, we obtain new proofs of the equivalences between different models of algebraic K-theory, given by the Q-, the oS-, the cobordism model, and the squares construction.
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BibTeX
@misc{raptis2026spansquares,
title = {{The span-squares adjunction}},
author = {George Raptis and Wolfgang Steimle},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.10052},
eprint = {2606.10052},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.10052},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The span-squares adjunction
AU - George Raptis
AU - Wolfgang Steimle
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2606.10052
UR - https://arxiv.org/abs/2606.10052
AB - We show a universal property of the span ∞-category that yields a description of functors defined on this category. For this, we view the span construction as a functor from double ∞-categories to ∞-categories, and show that this functor admits a right adjoint defined by the double ∞-categories of squares. As an application, we obtain a model for the localization of an ∞-category in terms of a span ∞-category and also give another proof of the model given by the relative nerve. Moreover, using the span-squares adjunction, we obtain new proofs of the equivalences between different models of algebraic K-theory, given by the Q-, the oS-, the cobordism model, and the squares construction.
ER -