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hypercomplete object

**(∞,1)-topos theory** ## Background ## * sheaf and topos theory * (∞,1)-category * (∞,1)-functor * (∞,1)-presheaf * (∞,1)-category of (∞,1)-presheaves ## Definitions ## * elementary (∞,1)-topos * (∞,1)-site * reflective sub-(∞,1)-category * localization of an (∞,1)-category * topological localization * hypercompletion * (∞,1)-category of (∞,1)-sheaves * (∞,1)-sheaf/∞-stack/derived stack * (∞,1)-topos * (n,1)-topos, n-topos * n-truncated object * n-connected object * (1,1)-topos * presheaf * sheaf * (2,1)-topos, 2-topos * (2,1)-presheaf * (∞,1)-quasitopos * separated (∞,1)-presheaf * quasitopos * separated presheaf * (2,1)-quasitopos? * separated (2,1)-presheaf * (∞,2)-topos * (∞,n)-topos ## Characterization ## * universal colimits * object classifier * groupoid object in an (∞,1)-topos * effective epimorphism ## Morphisms * (∞,1)-geometric morphism * (∞,1)Topos * Lawvere distribution ## Extra stuff, structure and property * hypercomplete (∞,1)-topos * hypercomplete object * Whitehead theorem * over-(∞,1)-topos * n-localic (∞,1)-topos * locally n-connected (n,1)-topos * structured (∞,1)-topos * geometry (for structured (∞,1)-toposes) * locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos * local (∞,1)-topos * concrete (∞,1)-sheaf * cohesive (∞,1)-topos ## Models ## * models for ∞-stack (∞,1)-toposes * model category * model structure on functors * model site/sSet-site * model structure on simplicial presheaves * descent for simplicial presheaves * descent for presheaves with values in strict ∞-groupoids ## Constructions ## **structures in a cohesive (∞,1)-topos** * shape / coshape * cohomology * homotopy * fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos/of a locally ∞-connected (∞,1)-topos * categorical/geometric homotopy groups * Postnikov tower * Whitehead tower * rational homotopy * dimension * homotopy dimension * cohomological dimension * covering dimension * Heyting dimension

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Contents

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 08:23:11 by Urs Schreiber (131.211.233.156)