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homotopy dimension

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Definition

Definition

An (∞,1)-topos 𝒳\mathcal{X} has homotopy dimension n\leq n \in \mathbb{N} if every (n-1)-connected object AA has a global element, a morphism *A* \to A from the terminal object into it.

This appears as HTT, def. 7.2.1.1.

Definition

An (∞,1)-topos 𝒳\mathcal{X} is locally of homotopy dimension n\leq n \in \mathbb{N} if there exists a collection {U i𝒳}\{U_i \in \mathcal{X}\} of objects such that

This appears as HTT, def. 7.2.1.8.

Properties

Proposition

If an (∞,1)-topos 𝒳\mathcal{X} is locally of homotopy dimension n\leq n for some nn \in \mathbb{N} then it is a hypercomplete (∞,1)-topos.

This appears as HTT, cor. 7.2.1.12.

Proposition

If 𝒳\mathcal{X} has homotopy dimension n\leq n then it also has cohomological dimension n\leq n.

The converse holds if 𝒳\mathcal{X} has finite homotopy dimension and n2n \geq 2.

This appears as HTT, cor. 7.2.2.30.

Proposition

An (∞,1)-topos 𝒳\mathcal{X} has homotopy dimension n\leq n precisely if the global section (∞,1)-geometric morphism

(ΔΓ):𝒳ΓΔGrpd (\Delta \dashv \Gamma) : \mathcal{X} \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd

has the property that Γ\Gamma sends (kn)(k\geq n)-connective morphisms to (kn)(k-n)-connective morphisms.

This is HTT, lemma 7.2.1.7

Examples

Proposition

Up to equivalence, the unique (∞,1)-topos of homotopy dimension 1\leq -1 is the the terminal category *Sh (,1)() * \simeq Sh_{(\infty,1)}(\emptyset).

This is HTT, example. 7.2.1.2.

Proof

An object X𝒳X \in \mathcal{X} is (1)(-1)-connected if the morphism X*X \to *to the terminal object in an (∞,1)-category is. This is the case if it is an effective epimorphism.

Since the global section (∞,1)-functor is corepresented by the terminal object, XX is 0-connective precisely if Γ(X)Γ(*)=*\Gamma(X) \to \Gamma(*) = * is an epimorphism on connected components. By the discussion at effective epimorphism, this is the case precisely if Γ(X)*\Gamma(X) \to * is an effective epimorphism in ∞Grpd.

So 𝒳\mathcal{X} has homotopy dimension 0\leq 0 if Γ\Gamma preserves effective epimorphisms. This is the case if it preserves finit (∞,1)-limits (the (∞,1)-pullbacks defining a Cech nerve) and all (∞,1)-colimits (over the resulting Cech nerve). being a right adjoint (∞,1)-functor Γ\Gamma always preserves (∞,1)-limits. If 𝒳\mathcal{X} is local then Γ\Gamma is by definition also a left adjoint and hence also preserves (∞,1)-colimits.

Proposition

Every local (∞,1)-topos has homotopy dimension 0\leq 0.

Proof

Let

(ΔΓ):HGrpd (\Delta \dashv \Gamma \dashv \nabla) : \mathbf{H} \to \infty Grpd

be the terminal geometric morphism of the local (,1)(\infty,1)-topos, with \nabla being the extra right adjoint to the global section (∞,1)-geometric morphism functor that characterizes locality.

By prop 3 it is sufficient to show that Γ\Gamma send (-1)-connected morphisms to (-1)-connected morphisms, hence effective epimorphisms to effective epimorphisms.

By the existence of \nabla we have that Γ\Gamma preserves not only (∞,1)-limits but also (∞,1)-colimits. Since effective epimorphisms are defined as certain colimits over diagrams of certain limits, Γ\Gamma preserves effective epimorphisms.

So in particular for CC any (∞,1)-category with a terminal object, the (∞,1)-category of (∞,1)-presheaves PSh (,1)(C)PSh_{(\infty,1)}(C) is an (∞,1)-topos of homotopy dimension 0\leq 0. Notably Top \simeq ∞Grpd PSh (,1)(*)\simeq PSh_{(\infty,1)}(*) has homotopy dimension 0\leq 0.

This is HTT, example. 7.2.1.3.

Proposition

Every (∞,1)-category of (∞,1)-presheaves is an (∞,1)-topos of local homotopy dimension 0\leq 0.

This appears as HTT, example. 7.2.1.9.

Theorem

If a paracompact topological space XX has covering dimension n\leq n, then the (∞,1)-category of (∞,1)-sheaves Sh (,1)(X):=Sh (,11)(Op(X))Sh_{(\infty,1)}(X) := Sh_{(\infty,11)}(Op(X)) is an (∞,1)-topos of homotopy dimension n\leq n.

This is HTT, theorem 7.2.3.6.

Proposition

For XX \in ∞Grpd \simeq Top an object, the over-(∞,1)-topos Grpd/X\infty Grpd/X has homotopy dimension n\leq n precisely if XTopX \in Top a retract in the homotopy category Ho(Top)Ho(Top) of a CW-complex of dimension n\leq n.

This is HTT, example 7.2.1.4.

References

The (∞,1)-topos theoretic notion is discuss in section 7.2.1 of

Revised on April 14, 2012 13:14:29 by Urs Schreiber (89.204.130.9)