(∞,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
This appears as HTT, def. 18.104.22.168.
An (∞,1)-topos is locally of homotopy dimension if there exists a collection of objects such that
This appears as HTT, def. 22.214.171.124.
This appears as HTT, cor. 126.96.36.199.
If has homotopy dimension then it also has cohomological dimension .
The converse holds if has finite homotopy dimension and .
This appears as HTT, cor. 188.8.131.52.
An (∞,1)-topos has homotopy dimension precisely if the global section (∞,1)-geometric morphism
has the property that sends -connective morphisms to -connective morphisms.
This is HTT, lemma 184.108.40.206
This is HTT, example. 220.127.116.11.
An object is -connected if the morphism to the terminal object in an (∞,1)-category is. This is the case if it is an effective epimorphism.
Since the global section (∞,1)-functor is corepresented by the terminal object, is 0-connective precisely if is an epimorphism on connected components. By the discussion at effective epimorphism, this is the case precisely if is an effective epimorphism in ∞Grpd.
So has homotopy dimension if preserves effective epimorphisms. This is the case if it preserves finit (∞,1)-limits (the (∞,1)-pullbacks defining a Cech nerve) and all (∞,1)-colimits (over the resulting Cech nerve). being a right adjoint (∞,1)-functor always preserves (∞,1)-limits. If is local then is by definition also a left adjoint and hence also preserves (∞,1)-colimits.
So in particular for any (∞,1)-category with a terminal object, the (∞,1)-category of (∞,1)-presheaves is an (∞,1)-topos of homotopy dimension . Notably Top ∞Grpd has homotopy dimension .
This is HTT, example. 18.104.22.168.
This appears as HTT, example. 22.214.171.124.
This is HTT, theorem 126.96.36.199.
This is HTT, example 188.8.131.52.
The (∞,1)-topos theoretic notion is discuss in section 7.2.1 of