André Joyal : "New variations on the notion of topos"

Abstract: The notion of topos is the proeminent member of a family of related notions which includes the notion of abelian category, of locally presentable category and that of higher topos. After discussing their common features, we shall propose the inclusions of two new members: the notion of locus and that of para-topos.

The category of pointed space and the category of spectra are examples of loci. The category of small (weak) n-categories  is an example of para-topos for every

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