Abstract: Let $X$ be a quasicategory. Then it is an $\infty$-topos, if it is an accessible left exact localization of the presheaf category of a small quasicategory. We will introduce two sets of intrinsic conditions which are equivalent to being an $\infty$-topos:

1) the $\infty$-categorical Giraud axioms, and

2) colimits in $X$ are universal, and it has small object classifiers for large enough regular cardinals,

and we discuss the equivalences.