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homotopy dimension

Definition

Definition

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

This appears as HTT, def. 7.2.1.1.

Definition

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

• the $\{U_i\}$ generate $𝒳$ under (∞,1)-colimits;

• each over-(∞,1)-topos $𝒳/U_i$ has homotopy dimension $\le n$.

This appears as HTT, def. 7.2.1.8.

Properties

Proposition

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

This appears as HTT, cor. 7.2.1.12.

Proposition

If $𝒳$ has homotopy dimension $\le n$ then it also has cohomological dimension $\le n$.

The converse holds if $𝒳$ has finite homotopy dimension and $n\ge 2$.

This appears as HTT, cor. 7.2.2.30.

Proposition

An (∞,1)-topos $𝒳$ has homotopy dimension $\le n$ precisely if the global section (∞,1)-geometric morphism

$$(\Delta ⊣\Gamma ):𝒳\stackrel{\stackrel{\Delta }{\leftarrow }}{\underset{\Gamma }{\to }}\mathrm{\infty }\mathrm{Grpd}$$(\Delta \dashv \Gamma) : \mathcal{X} \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd

has the property that $\Gamma$ sends $(k\ge n)$-connective morphisms to $(k-n)$-connective morphisms.

This is HTT, lemma 7.2.1.7

Examples

Proposition

Up to equivalence, the unique (∞,1)-topos of homotopy dimension $\le -1$ is the the terminal category $*\simeq {\mathrm{Sh}}_{(\mathrm{\infty },1)}(\varnothing )$.

This is HTT, example. 7.2.1.2.

Proof

An object $X\in 𝒳$ is $(-1)$-connected if the morphism $X\to *$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, $X$ is 0-connective precisely if $\Gamma (X)\to \Gamma (*)=*$ is an epimorphism on connected components. By the discussion at effective epimorphism, this is the case precisely if $\Gamma (X)\to *$ is an effective epimorphism in ∞Grpd.

So $𝒳$ has homotopy dimension $\le 0$ if $\Gamma$ 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 $\Gamma$ always preserves (∞,1)-limits. If $𝒳$ is local then $\Gamma$ is by definition also a left adjoint and hence also preserves (∞,1)-colimits.

Proposition

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

Proof

Let

$$(\Delta ⊣\Gamma ⊣\nabla ):H\to \mathrm{\infty }\mathrm{Grpd}$$(\Delta \dashv \Gamma \dashv \nabla) : \mathbf{H} \to \infty Grpd

be the terminal geometric morphism of the local $(\mathrm{\infty },1)$-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 $\Gamma$ send (-1)-connected morphisms to (-1)-connected morphisms, hence effective epimorphisms to effective epimorphisms.

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

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

This is HTT, example. 7.2.1.3.

Proposition

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

This appears as HTT, example. 7.2.1.9.

Theorem

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

This is HTT, theorem 7.2.3.6.

Proposition

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

This is HTT, example 7.2.1.4.

Related concepts

• dimension

  • homotopy dimension

  • cohomological dimension

  • covering dimension

  • Heyting dimension

References

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

• Jacob Lurie, Higher Topos Theory