nLab
hypercomplete object
**
(∞,1)-topos theory **
## Background ##
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sheaf and topos theory
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(∞,1)-category
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(∞,1)-functor
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(∞,1)-presheaf
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(∞,1)-category of (∞,1)-presheaves
## Definitions ##
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elementary (∞,1)-topos
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(∞,1)-site
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reflective sub-(∞,1)-category
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localization of an (∞,1)-category
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topological localization
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hypercompletion
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(∞,1)-category of (∞,1)-sheaves
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(∞,1)-sheaf /
∞-stack /
derived stack
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(∞,1)-topos
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(n,1)-topos ,
n-topos
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n-truncated object
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n-connected object
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(1,1)-topos
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presheaf
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sheaf
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(2,1)-topos ,
2-topos
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(2,1)-presheaf
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(∞,1)-quasitopos
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separated (∞,1)-presheaf
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quasitopos
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separated presheaf
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(2,1)-quasitopos?
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separated (2,1)-presheaf
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(∞,2)-topos
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(∞,n)-topos
## Characterization ##
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universal colimits
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object classifier
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groupoid object in an (∞,1)-topos
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effective epimorphism
## Morphisms
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(∞,1)-geometric morphism
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(∞,1)Topos
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Lawvere distribution
## Extra stuff, structure and property
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hypercomplete (∞,1)-topos
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hypercomplete object
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Whitehead theorem
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over-(∞,1)-topos
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n-localic (∞,1)-topos
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locally n-connected (n,1)-topos
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structured (∞,1)-topos
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geometry (for structured (∞,1)-toposes)
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locally ∞-connected (∞,1)-topos ,
∞-connected (∞,1)-topos
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local (∞,1)-topos
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concrete (∞,1)-sheaf
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cohesive (∞,1)-topos
## Models ##
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models for ∞-stack (∞,1)-toposes
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model category
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model structure on functors
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model site /
sSet-site
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model structure on simplicial presheaves
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descent for simplicial presheaves
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descent for presheaves with values in strict ∞-groupoids
## Constructions ##
**structures in a
cohesive (∞,1)-topos **
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shape /
coshape
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cohomology
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homotopy
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fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos /
of a locally ∞-connected (∞,1)-topos
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categorical /
geometric homotopy groups
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Postnikov tower
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Whitehead tower
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rational homotopy
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dimension
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homotopy dimension
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cohomological dimension
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covering dimension
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Heyting dimension
Contents
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 equivalence s 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 morphism s.
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 fiber s 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.
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
The definition appears before lemma 6.5.2.9
Created on May 14, 2010 08:23:11
by
Urs Schreiber
(131.211.233.156)