# nLab homotopy dimension

### Context

#### $\left(\infty ,1\right)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Definition

###### Definition

An (∞,1)-topos $𝒳$ has homotopy dimension $\le n\in ℕ$ 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 $\le n\in ℕ$ if there exists a collection $\left\{{U}_{i}\in 𝒳\right\}$ of objects such that

• the $\left\{{U}_{i}\right\}$ 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 $n\in ℕ$ 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

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

has the property that $\Gamma$ sends $\left(k\ge n\right)$-connective morphisms to $\left(k-n\right)$-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 $*\simeq {\mathrm{Sh}}_{\left(\infty ,1\right)}\left(\varnothing \right)$.

This is HTT, example. 7.2.1.2.

###### Proof

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

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

be the terminal geometric morphism of the local $\left(\infty ,1\right)$-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 ${\mathrm{PSh}}_{\left(\infty ,1\right)}\left(C\right)$ is an (∞,1)-topos of homotopy dimension $\le 0$. Notably Top $\simeq$ ∞Grpd $\simeq {\mathrm{PSh}}_{\left(\infty ,1\right)}\left(*\right)$ 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 ${\mathrm{Sh}}_{\left(\infty ,1\right)}\left(X\right):={\mathrm{Sh}}_{\left(\infty ,11\right)}\left(\mathrm{Op}\left(X\right)\right)$ 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 $\infty \mathrm{Grpd}/X$ has homotopy dimension $\le n$ precisely if $X\in \mathrm{Top}$ a retract in the homotopy category $\mathrm{Ho}\left(\mathrm{Top}\right)$ of a CW-complex of dimension $\le n$.

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)