(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
A classical theorem of Giraud characterizes sheaf toposes abstractly as categories with certain properties known as Giraud’s axioms.
In higher topos theory there are corresponding higher analogs .
(Giraud’s theorem)
A category is a Grothendieck topos if and only if it satisfies the following conditions:
it is locally presentable;
it has universal colimits;
it has disjoint coproducts;
it has effective quotients.
Similarly:
(Giraud-Rezk-Lurie theorem)
An $(\infty,1)$-category is an Grothendieck-Rezk-Lurie-(infinity,1)-topos if and only if it satisfies the following conditions:
it is locally presentable;
it has universal $\infty$-colimits;
it has disjoint coproducts;
it has effective groupoid objects.
Similarly, in 2-category theory there is the analogous 2-Giraud theorem for Grothendieck 2-toposes.
Textbook accounts:
following:
Lecture notes:
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