equivalences in/of $(\infty,1)$-categories
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
An (∞,1)-category $C$ is an $(\infty,1)$-semitopos if
it is presentable (∞,1)-category;
it has universal colimits;
for every morphism the corresponding Cech nerve groupoid object is effective.
This appears as HTT, def. 6.2.3.1.
If $C$ is an $(\infty,1)$-semitopos and $X \in C$ is any object, then also the over-(∞,1)-category is an $(\infty,1)$-semitopos.
This appears as (HTT, remark 6.2.3.3).
semitopos?