(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
An object in an (∞,1)-topos is hypercomplete if it regards the Whitehead theorem to be true in , i.e. if homming weak homotopy equivalences into it produces an equivalence.
Let be an (∞,1)-topos.
An object is hypercomplete if it is a local object with respect to all -connected morphisms.
This means: if for every morphism which is -connected as an object of the over category (roughly: all its homotopy fibers have vanishing homotopy groups), then the induced morphism
is an equivalence in a quasi-category in ∞Grpd.
The -topos itself is a hypercomplete (∞,1)-topos if all its objects are hyercomplete. See there for more details.
Hypercompleteness is a notion that appears only due to the possible unboundedness of the degree of homotopy groups in an (∞,1)-topos. The notion is empty in an (n,1)-topos for finite .
An object being hypercomplete in means that it regards the Whitehead theorem to be true in . If itself is hypercomplete, then the Whitehead theorem is true in .
This is the topic of section 6.5.2 of
The definition appears before lemma 6.5.2.9
Created on May 14, 2010 at 07:51:32. See the history of this page for a list of all contributions to it.