Year: 2026
Type: preprint
Abstract: We show a universal property of the span $\infty$-category that yields a description of functors defined on this category. For this, we view the span construction as a functor from double $\infty$-categories to $\infty$-categories, and show that this functor admits a right adjoint defined by the double $\infty$-categories of squares. As an application, we obtain a model for the localization of an $\infty$-category in terms of a span $\infty$-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.
Source: arXiv (Cornell University)
Authors: Raptis, George, Steimle, Wolfgang
Institutions: University of Augsburg, Aristotle University of Thessaloniki
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7164150641