Contents

Idea

An object in an (∞,1)-topos $H$ is hypercomplete if it regards the Whitehead theorem to be true in $H$, i.e. if homming weak homotopy equivalences into it produces an equivalence.

Definition

Let $H$ be an (∞,1)-topos.

An object $A\in H$ is hypercomplete if it is a local object with respect to all $\mathrm{\infty }$-connected morphisms.

This means: if for every morphism $f:X\to Y$ which is $\mathrm{\infty }$-connected as an object of the over category $H_{/Y}$ (roughly: all its homotopy fibers have vanishing homotopy groups), then the induced morphism

is an equivalence in a quasi-category in ∞Grpd.

The $(\mathrm{\infty },1)$-topos $H$ itself is a hypercomplete (∞,1)-topos if all its objects are hyercomplete. See there for more details.

Remarks

• 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 $n$.

• An object being hypercomplete in $H$ means that it regards the Whitehead theorem to be true in $H$. If $H$ itself is hypercomplete, then the Whitehead theorem is true in $H$.

References

This is the topic of section 6.5.2 of

• Jacob Lurie, Higher Topos Theory

The definition appears before lemma 6.5.2.9