# nLab hypercomplete object

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# 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 $\infty$-connected morphisms.

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

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

The $\left(\infty ,1\right)$-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

The definition appears before lemma 6.5.2.9

Created on May 14, 2010 08:23:11 by Urs Schreiber (131.211.233.156)