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

between (∞,1)-toposes such that that:

• $F$ is generated by the image of $E$ under (∞,1)-colimits;

• the right adjoint of$j_!$right adjoint of $j_!$ preserves (∞,1)-colimits;

• for every $U\in E$ , the induced left adjointleft adjoint

  $$(j_{!}{)}_{/U}:E_{/U}<span><del\; class="diffmod">\; \to </del><ins\; class="diffmod">\; ⟶</ins></span>F_{/j_{!}U}$$ (j_!)_{/U}\colon E_{/U}\to (j_!)_{/U} \;\colon\; E_{/U} \longrightarrow F_{/j_!U}

  is fully faithful;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

• fractured homotopy type theory

  big and little toposes

• fractured homotopy type theory

References

• André Joyal ,

  André Joyal, Ieke Moerdijk: A completeness theorem for open maps, Annals of Pure and Applied Logic 70 1 (1994) 51–86 [doi:10.1016/0168-0072(94)90069-8]

  Ieke Moerdijk , A completeness theorem for open maps, Annals of Pure and Applied Logic 70:1 (1994), 51–86. JMdoi

• Eduardo J. Dubuc, Axiomatic etal maps and a theory of spectrum, Journal of Pure and Applied Algebra 149 1 (2000) 15–45 [doi:10.1016/s0022-4049(98)00161-3]

• 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]

• 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), pdf]

• 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), pdf]

• Nima Rasekh, What is a topological structure?, talk notes (April 2023) [[pdf](https://guests.mpim-bonn.mpg.de/rasekh/Condensed.pdf), pdf]