+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Cohesive $\infty$-Toposes +--{: .hide} [[!include cohesive infinity-toposes - contents]] =-- =-- =-- \tableofcontents ## Idea Roughly, fractured (∞,1)-toposes axiomatize the notion of an [[orbifold]] (or, rather, [[etale stack]]) and [[etale map]] in any [[(∞,1)-topos]]. The 1-categorical version is due to Joyal–Moerdijk and Dubuc and the (∞,1)-categorical version is due to Carchedi and Lurie. ## Definition A __fractured (∞,1)-topos__ is a [[left adjoint]] [[(∞,1)-functor]] $$ j_! \;\colon\; E \longrightarrow F $$ between [[(∞,1)-toposes]] such that: * $F$ is generated by the image of $E$ under [[(∞,1)-colimits]]; * the [[right adjoint]] of $j_!$ preserves [[(∞,1)-colimits]]; * for every $U\in E$, the induced [[left adjoint]] $$ (j_!)_{/U} \;\colon\; E_{/U} \longrightarrow F_{/j_!U} $$ is [[fully faithful (infinity,1)-functor|fully faithful]]; * maps in the image of $j_!$ are stable under base changes along maps with domain in the image of $j_!$. ## Example Take $F$ to be the [[(∞,1)-topos]] of sheaves on the [[site of cartesian spaces]] and smooth maps and $E$ to be the (∞,1)-topos of sheaves on the site of cartesian spaces and open embeddings. Take $j_!$ to be the unique (∞,1)-cocontinuous functor induced by the corresponding inclusion of sites. Objects in the image of $j_!$ are known as __etale stacks__. Maps in the image of $j_!$ are known as __etale maps__. ## Applications For any object $U\in E$, we define the __[[petit topos]]__ of $U$ as $E_{/U}$ and the __[[gros topos]]__ of $U$ as $F_{/j_! U}$. ## Examples * Apparently [[condensed mathematics]] takes place in fractured $\infty$-topoi, see at *[[condensed local contractibility]]*. ## Related concepts * [[big and little toposes]] * [[fractured homotopy type theory]] ## References * [[André Joyal]], [[Ieke Moerdijk]]: _A completeness theorem for open maps_, Annals of Pure and Applied Logic **70** 1 (1994) 51–86 \[<a href="https://doi.org/10.1016/0168-0072(94)90069-8">doi:10.1016/0168-0072(94)90069-8</a>\] * [[Eduardo J. Dubuc]], _Axiomatic etal maps and a theory of spectrum_, Journal of Pure and Applied Algebra **149** 1 (2000) 15–45 \[<a href="https://doi.org/10.1016/s0022-4049(98)00161-3">doi:10.1016/s0022-4049(98)00161-3</a>\] * [[David Carchedi]]: *Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity Topoi*, Memoirs of the American Mathematical Society **264** 1282 (2020) [[doi:10.1090/memo/1282](https://doi.org/10.1090/memo/1282), [arXiv:1312.2204](https://arxiv.org/abs/1312.2204)] * [[Jacob Lurie]]: *Fractured $\infty$-Topoi*, Chapter 20 in: *[[Spectral Algebraic Geometry]]* (2018) The example of [[condensed local contractibility]]: * [[Qi Zhu]], *Fractured Structure on Condensed Anima*, MSc thesis (2023) [[pdf](https://1429cecd-24a0-4223-8b7c-1ebf47aa12e2.filesusr.com/ugd/8e912a_fe1c2f8e90094d0fa065fb8129687963.pdf), [[QiZhu-FracturedCondensed.pdf:file]]] * [[Nima Rasekh]], [[Qi Zhu]], *Fractured Structures in Condensed Mathematics* [[arXiv:2603.09618](https://arxiv.org/abs/2603.09618)] exposition: * [[Qi Zhu]], *Fractured structure on condensed spaces*, talk notes (2023) [[pdf](https://1429cecd-24a0-4223-8b7c-1ebf47aa12e2.filesusr.com/ugd/8e912a_8d2a90176ed14969a877e77ec1b787da.pdf), [[QiZhuFracturedCondensed.pdf:file]]] * [[Nima Rasekh]], *What is a topological structure?*, talk notes (April 2023) [[pdf](https://guests.mpim-bonn.mpg.de/rasekh/Condensed.pdf), [[Rasekh-TopologicalStructure.pdf:file]]] [[!redirects fractured (infinity,1)-toposes]] [[!redirects fractured (infinity,1)-topoi]] [[!redirects fractured (∞,1)-topos]] [[!redirects fractured (∞,1)-toposes]] [[!redirects fractured (∞,1)-topoi]] [[!redirects fractured topos]] [[!redirects fractured toposes]] [[!redirects fractured topoi]]