(∞,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 notion of -quasitopos is the (∞,1)-topos-analog of the notion of quasitopos.
Let be an (∞,1)-bisite. Say an (∞,1)-presheaf is -biseparated if it is an (∞,1)-sheaf for and for every -covering sieve in we have that the induced morphism
in ∞Grpd is a full and faithful (∞,1)-functor.
We say it is -biseparated if
the induced morphism
is an (n-1)-truncated object in the (∞,1)-overcategory .
A (Grothendieck) -quasitopos is an (∞,1)-category that is equivalent to the full sub-(∞,1)-category of some on the -biseparated -presheaves, on some (∞,1)-bisite .
For a local (∞,1)-topos
and be a site of definition for , the -quasitopos on that factors the geometric embedding
is that of concrete objects in , the analog of concrete sheaves.
The definition as it stands, originated out of a discussion between Urs Schreiber and David Carchedi. The suggestion to rephrase the definition in terms of bisites came from Mike Shulman.