equivalences in/of $(\infty,1)$-categories
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
The theory of (∞,1)-toposes, generalizing topos theory from category theory to (∞,1)-category theory: “geometric homotopy theory”.
$(\infty,1)$-topos theory
An quick introduction is in part 3, 4 of
For origins of the notion of $(\infty,1)$-topos itself see the references at (∞,1)-topos.
Early frameworks for Grothendieck (as opposed to “elementary”) $(\infty,1)$-topoi are due Charles Rezk via model categories
and due to Toën–Vezzosi in two versions (preprints 2002), via simplically enriched categories and via Segal categories:
Bertrand Toën, Gabriele Vezzosi, Homotopical algebraic geometry. I. Topos theory, Adv. Math. 193 (2005) no. 2, 257–372 doi arXiv:math.AT/0207028
Bertrand Toën , Gabriele Vezzosi, Segal topoi and stacks over Segal categories, arXiv:math.AG/0212330
A general abstract conception of $(\infty,1)$-topos theory in terms of (∞,1)-category theory was given in
The analog of the Elephant for $(\infty,1)$-topos theory is still to be written.