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A couple of people are working together on relating condensed mathematics to a kind of local cohesiveness. They speak of the concept of a fractured structure/fractured topos, presumably related to fracture squares.
Their talk notes: Qi Zhu, Fractured structure on condensed spaces; Nima Rasekh, What is a topological structure?.
They are referring to the notion of “fractured $\infty$-topos” in Ch. 20 of Spectral Algebraic Geometry (pp. 1498).
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