Higher category theory
higher category theory
Extra properties and structure
Higher topos Theory
(∞,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
An -sheaf or -stack is the higher analog of an (∞,1)-sheaf / ∞-stack.
For an (∞,1)-category equipped with the structure of an (∞,1)-site, an -sheaf on is an (∞,1)-functor
to (∞,1)Cat, that satisfies descent: hence which is a local object with respect to the covering sieve inclusions in .
The (∞,2)-category of -sheaves
is an (∞,2)-topos, the homotopy theory-generalization of a 2-topos of 2-sheaves.
Codomain fibration / canonical -sheaf
Let be an (∞,1)-topos, regarded as a (large) (∞,1)-site equipped with the canonical topology. Then an (∞,1)-functor
is an -sheaf precisely if it preserves (∞,1)-limits (takes (∞,1)-colimits in to (∞,1)-limits in (∞,1)Cat).
For an -topos, the functor
is a (large) -sheaf on , regarded as a (∞,1)-site equipped with the canonical topology. Here is the slice (∞,1)-topos over .
This is a special case of (Lurie, lemma 126.96.36.199).
Discussion of a local model structure on simplicial presheaves with respect to the Joyal model structure for quasicategories is in