nLab fractured (infinity,1)-topos

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;

• 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 [doi:10.1016/0168-0072(94)90069-8]

• 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, 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, pdf]

• Nima Rasekh, Qi Zhu, Fractured Structures in Condensed Mathematics [arXiv:2603.09618]

exposition:

• Qi Zhu, Fractured structure on condensed spaces, talk notes (2023) [pdf, pdf]

• Nima Rasekh, What is a topological structure?, talk notes (April 2023) [pdf, pdf]