nLab hypercomplete object

Idea

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

Definition

Let $\mathbf{H}$ be an (∞,1)-topos.

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

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

$$\mathbf{H}(f,A) : \mathbf{H}(Y,A) \to \mathbf{H}(X,A)$$

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

The $(\infty,1)$-topos $\mathbf{H}$ 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 $\mathbf{H}$ means that it regards the Whitehead theorem to be true in $\mathbf{H}$. If $\mathbf{H}$ itself is hypercomplete, then the Whitehead theorem is true in $\mathbf{H}$.

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