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hypercomplete object

(∞,1)-topos theory

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Idea

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

Definition

Let H\mathbf{H} be an (∞,1)-topos.

An object AHA \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:XYf : X \to Y which is \infty-connected as an object of the over category H /Y\mathbf{H}_{/Y} (roughly: all its homotopy fibers have vanishing homotopy groups), then the induced morphism

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

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

The (,1)(\infty,1)-topos H\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 nn.

  • An object being hypercomplete in H\mathbf{H} means that it regards the Whitehead theorem to be true in H\mathbf{H}. If H\mathbf{H} itself is hypercomplete, then the Whitehead theorem is true in H\mathbf{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 at 08:23:11. See the history of this page for a list of all contributions to it.