Unfolding of symmetric monoidal (∞,n)-categories
Abstract
We prove Lurie's proposed description of left rigid symmetric monoidal (∞,n)-categories in terms of purely ∞-categorical data. More precisely, unfolding gives an equivalence between left rigid symmetric monoidal (∞,n)-categories and complete categorical chain complexes of rigid symmetric monoidal ∞-categories. More generally, unfolding identifies skeletal sequences with categorical chain complexes without the completeness condition. We extend these equivalences to (∞,∞)-categories and, stably, obtain an equivalence between complete unbounded categorical chain complexes and left rigid categorical spectra. We also analyze their groupoid completion and express the homotopy groups of the associated spaces and spectra in terms of categorical cycles and boundaries. As examples, we construct the categorical chain complexes associated to geometric and algebraic bordism categories, including their equivariant variants, and to iterated span categories. Their localizations recover classical bordism and L-groups, while the span examples are related to algebraic K-theory.
Related
- On closed symmetric monoidal enriched ∞-categories
- The symmetric skew monoidal structure of Gray-categories
- On pushouts of strict symmetric monoidal categories
- Monoidal Categories associated with Kac-Moody Open Richardson Varieties in Symmetric Type
- Quantum Hopf rigidity in symmetric monoidal categories
BibTeX
@misc{haugseng2026unfolding,
title = {{Unfolding of symmetric monoidal $(\infty,n)$-categories}},
author = {Rune Haugseng and Thomas Nikolaus},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01994},
eprint = {2611.01994},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01994},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Unfolding of symmetric monoidal (∞,n)-categories
AU - Rune Haugseng
AU - Thomas Nikolaus
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01994
UR - https://arxiv.org/abs/2611.01994
AB - We prove Lurie's proposed description of left rigid symmetric monoidal (∞,n)-categories in terms of purely ∞-categorical data. More precisely, unfolding gives an equivalence between left rigid symmetric monoidal (∞,n)-categories and complete categorical chain complexes of rigid symmetric monoidal ∞-categories. More generally, unfolding identifies skeletal sequences with categorical chain complexes without the completeness condition. We extend these equivalences to (∞,∞)-categories and, stably, obtain an equivalence between complete unbounded categorical chain complexes and left rigid categorical spectra. We also analyze their groupoid completion and express the homotopy groups of the associated spaces and spectra in terms of categorical cycles and boundaries. As examples, we construct the categorical chain complexes associated to geometric and algebraic bordism categories, including their equivariant variants, and to iterated span categories. Their localizations recover classical bordism and L-groups, while the span examples are related to algebraic K-theory.
ER -