The (∞,∞)-category of spans
Abstract
In this paper, we construct the (∞,∞)-category Span∞(𝒞) of spans, also known as correspondences, in any given (∞,1)-category 𝒞 with finite limits. This yields new models for the span (∞,n)-categories for n ∈ ℕ ∪ {∞}. We characterize the mapping (∞, n−1)-categories in these (∞,n)-categories, and thereby verify that our model agrees with other models for spans. Finally, and most importantly, we prove a new universal property, characterizing functors into span (∞,n)-categories, which specializes to the well-known relation with the twisted arrow categories in dimension $1$. These results will be used in the sequels to construct higher analogs of the classical Hall algebra construction, where "higher" refers to both higher categorical and "higher monoidal" structures, i.e., Ek-algebras in (∞, n)-categories for n, k>1.
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BibTeX
@misc{godicke2026category,
title = {{The $(\infty,\infty)$-category of spans}},
author = {Jonte Gödicke and Quoc P. Ho and Walker H. Stern},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29495},
eprint = {2608.29495},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29495},
}
RIS
TY - UNPB
TI - The (∞,∞)-category of spans
AU - Jonte Gödicke
AU - Quoc P. Ho
AU - Walker H. Stern
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29495
UR - https://arxiv.org/abs/2608.29495
AB - In this paper, we construct the (∞,∞)-category Span_∞(𝒞) of spans, also known as correspondences, in any given (∞,1)-category 𝒞 with finite limits. This yields new models for the span (∞,n)-categories for n ∈ ℕ ∪ {∞}. We characterize the mapping (∞, n−1)-categories in these (∞,n)-categories, and thereby verify that our model agrees with other models for spans. Finally, and most importantly, we prove a new universal property, characterizing functors into span (∞,n)-categories, which specializes to the well-known relation with the twisted arrow categories in dimension $1$. These results will be used in the sequels to construct higher analogs of the classical Hall algebra construction, where "higher" refers to both higher categorical and "higher monoidal" structures, i.e., E_k-algebras in (∞, n)-categories for n, k>1.
ER -