(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
The theory of (∞,1)-toposes, generalizing topos theory from category theory to (∞,1)-category theory.
For origins of the notion of -topos itself see the references at (∞,1)-topos.
Early frameworks for Grothendieck (as opposed to “elementary”) -topoi are due Charles Rezk and Toën–Vezzosi in two versions (preprints 2002), via simplically enriched categories and via Segal categories:
A general abstract conception of -topos theory in terms of (∞,1)-category theory was given in
The analog of the Elephant for -topos theory is still to be written.