Left-exact localizations of ∞-topoi [ABFJ]
注意同样四位作者还有另外一篇文章, 见 ABFJ-BM.
Left-exact localizations of ∞-topoi I: Higher sheaves
@article{ANEL2022108268,
title = {Left-exact localizations of ∞-topoi I: Higher sheaves},
journal = {Advances in Mathematics},
volume = {400},
pages = {108268},
year = {2022},
issn = {0001-8708},
doi = {https://doi.org/10.1016/j.aim.2022.108268},
url = {https://www.sciencedirect.com/science/article/pii/S0001870822000846},
author = {Mathieu Anel and Georg Biedermann and Eric Finster and André Joyal},
keywords = {Infinity-topos, Left-exact localization, Sheaf, Site, Congruence, Acyclic class},
abstract = {We are developing tools for working with arbitrary left-exact localizations of ∞-topoi. We introduce a notion of higher sheaf with respect to an arbitrary set of maps Σ in an ∞-topos E. We show that the full subcategory of higher sheaves Sh(E,Σ) is an ∞-topos, and that the sheaf reflection E→Sh(E,Σ) is the left-exact localization generated by Σ. The proof depends on the notion of congruence, which is a substitute for the notion of Grothendieck topology in 1-topos theory.}
}
Left-exact localizations of ∞-topoi II: Grothendieck topologies
@article{ANEL2024107472,
title = {Left-exact localizations of ∞-topoi II: Grothendieck topologies},
journal = {Journal of Pure and Applied Algebra},
volume = {228},
number = {3},
pages = {107472},
year = {2024},
issn = {0022-4049},
doi = {https://doi.org/10.1016/j.jpaa.2023.107472},
url = {https://www.sciencedirect.com/science/article/pii/S0022404923001548},
author = {Mathieu Anel and Georg Biedermann and Eric Finster and André Joyal},
keywords = {∞-topos, Grothendieck topologies, Hypercompletion, Acyclic classes},
abstract = {We revisit the work of Toën–Vezzosi and Lurie on Grothendieck topologies, using the new tools of acyclic classes and congruences. We introduce a notion of extended Grothendieck topology on any ∞-topos, and prove that the poset of extended Grothendieck topologies is isomorphic to that of topological localizations, hypercomplete localizations, Lawvere–Tierney topologies, and covering topologies (a variation on the notion of pretopology). It follows that these posets are small and have the structure of a frame. We revisit also the topological–cotopological factorization by introducing the notion of a cotopological morphism. And we revisit the notions of hypercompletion, hyperdescent, hypercoverings and hypersheaves associated to an extended Grothendieck topology. We also introduce the notion of forcing, which is a tool to compute with localizations of ∞-topoi. We use this in particular to show that the topological part of a left-exact localization of an ∞-topos is universally forcing the generators of this localization to be ∞-connected instead of inverting them.}
}
Left-exact Localizations of -Topoi III: The Acyclic Product
@misc{anel2025leftexactlocalizationsinftytopoiiii,
title={Left-exact Localizations of $\infty$-Topoi III: The Acyclic Product},
author={Mathieu Anel and Georg Biedermann and Eric Finster and André Joyal},
year={2025},
eprint={2308.15573},
archivePrefix={arXiv},
primaryClass={math.CT},
url={https://arxiv.org/abs/2308.15573},
}