nLab
n-connected object of an (infinity,1)-topos

Idea

An $n$-connected object is an object all whose homotopy groups equal to or below degree $n$ are trivial.

More precisely, an object in an ∞-stack (∞,1)-topos is $n$-connected if its categorical homotopy groups equal to or below degree $n$ are trivial.

The complementary notion is that of an n-truncated object of an (∞,1)-category.

The Whitehead tower construction produces $n$-connected objects.

Definition

Definition

An object $X$ in an (∞,1)-topos $H$ is called $n$-connected for $-1\le n\in \mathbb{Z}$ if

1. the terminal morphism $X\to *$ is an effective epimorphism;

2. all categorical homotopy groups equal to or below degree $n$ are trivial.

   $${\pi }_k(X)=*\mathrm{for}k\le n.$$\pi_k(X) = * \;\;\; for k \leq n \,.

A morphism $f:X\to Y$ in an $(\mathrm{\infty },1)$-topos is called $n$-connected if

1. it is an effective epimorphism in an (∞,1)-category

2. regarded as an object in the over-(∞,1)-category $H_{/Y}$ all categorical homotopy groups equal to or below degree $n$ are trivial.

This appears as HTT, def. 6.5.1.10, but under the name ”$(n+1)$-connective”. Another possible term is ”$n$-simply connected”; see n-connected space for discussion.

One adopts the following convenient terminology.

• Every object is $(-2)$-connected.

• A $(-1)$-connected object is also called an inhabited object.

• A 0-connected object is simply called a connected object.

Notice that effective epimorphisms are precisely the $(-1)$-connected morphisms. For more on this see n-connected/n-truncated factorization system.

Properties

General

Proposition

An object $X$ is $n$-connected (for $n\ge -2$) precisely if its n-truncation ${\tau }_{\le n}X$ is the terminal object of $H$.

This is HTT, prop. 6.5.1.12.

Observation

Every equivalence is $\mathrm{\infty }$-connected.

This is HTT, prop. 6.5.1.16, item 2.

Remark

In a general $(\mathrm{\infty },1)$-topos the converse is not true: not every $\mathrm{\infty }$-connected morphisms needs to be an equivalence. It is true in a hypercomplete (∞,1)-topos.

Proposition

The class of $n$-connected morphisms is stable under pullback and pushout.

If the pullback of a morphism along an effective epimorphism is $n$-connected, then so is the original morphism.

This is HTT, prop. 6.5.1.16, item 6.

Recursive characterization

Proposition

A morphism $f:X\to Y$ is $n$-connected precisely if it is an effective epimorphism and the diagonal morphism into the (∞,1)-pullback

$${\Delta }_f:X\to X{\times }_{Y}X$$\Delta_f : X \to X \times_Y X

is $(n-1)$-connected.

This appears as HTT, prop. 6.5.1.18.

Factorization system

Proposition

Let $H$ be an (∞,1)-topos. For all $(-2)\le n\le \mathrm{\infty }$ the class of $n$-connected morphisms in $H$ forms the left class in a orthogonal factorization system in an (∞,1)-category. The right class is that of n-truncated morphisms in $H$.

See also n-connected/n-truncated factorization system.

This appears as a remark in HTT, Example 5.2.8.16. A construction of the factorization in terms of a model category presentation is in (Rezk, prop. 8.5).

The truncated / connected clock

In a hypercomplete (∞,1)-topos the $\mathrm{\infty }$-connected morphisms are precisely the equivalences.

Therefore in such a context we have the following “clock” of notions of truncated object in an (infinity,1)-category / connected :

• any morphism = $(-2)$-connected

• effective epimorphism = $(-1)$-connected

• 0-connected, 1-connected, 2-connected, $\cdots$;

• $\mathrm{\infty }$-connected = equivalence = $(-2)$-truncated

• monomorphism = $(-1)$-truncated

• 0-truncated, 1-truncated, 2-truncated, $\cdots$

• $\mathrm{\infty }$-truncated = any morphism

Examples

In $\mathrm{Top}$

In the the (∞,1)-category Top we have that an object is $n$-connected precisely if it is an n-connected topological space:

• a $(-1)$-connected object is an inhabited space.

• a $0$-connected object is a path-connected space.

• a $1$-connected object is a simply connected space.

• a $\mathrm{\infty }$-connected object is a contractible space.

In $\mathrm{Grpd}$

Proposition

Let $f:X\to Y$ be a functor between groupoids. Regarded as a morphism in ∞Grpd $f$ is 0-connected precisely if it is an essentially surjective and full functor.

Proof

As discussed there, an effective epimorphism in ∞Grpd between 1-groupoids is precisely an essentially surjective functor.

So it remains to check that for an essentially surjective $f$, being 0-connected is equivalent to being full.

The homotopy pullback $X{\times }_{Y}X$ is given by the groupoid whose objects are triples $(x_1,x_2\in X,\alpha :f(x_1)\to f(x_2))$ and whose morphisms are corresponding tuples of morphisms in $X$ making the evident square in $Y$ commute.

By prop. 3 it is sufficient to check that the diagonal functor $X\to X{\times }_{Y}X$ is (-1)-connected, hence, as before, essentially surjective, precisely if $f$ is full.

First assume that $f$ is full. Then for $(x_1,x_2,\alpha )\in X{\times }_{Y}X$ any object, by fullness of $f$ there is a morphism $\hat{\alpha }:x_1\to x_2$ in $X$, such that $f(\hat{\alpha })=\alpha$.

Accordingly we have a morphism $(\hat{\alpha },\mathrm{id}):(x_1,x_2)\to (x_2,x_2)$ in $X{\times }_{Y}X$

$$\begin{array}{ccc}f(x_1)& \stackrel{f(\hat{\alpha })}{\to }& f(x_2)\\ {\downarrow }^{\alpha }& & {\downarrow }^{\mathrm{id}}\\ f(x_2)& \stackrel{\mathrm{id}}{\to }& f(x_2)\end{array}$$\array{ f(x_1) &\stackrel{f(\hat \alpha)}{\to}& f(x_2) \\ \downarrow^{\mathrlap{\alpha}} && \downarrow^{\mathrlap{id}} \\ f(x_2) &\stackrel{id}{\to}& f(x_2) }

to an object in the diagonal.

Conversely, assume that the diagonal is essentially surjective. Then for every pair of objects $x_1,x_2\in X$ such that there is a morphism $\alpha :f(x_1)\to f(x_2)$ we are guaranteed morphisms $h_1:x_1\to x_2$ and $h_2:x_2\to x_2$ such that

$$\begin{array}{ccc}f(x_1)& \stackrel{f(h_1)}{\to }& f(x_2)\\ {\downarrow }^{\alpha }& & {\downarrow }^{\mathrm{id}}\\ f(x_2)& \stackrel{f(h_2)}{\to }& f(x_2)\end{array}.$$\array{ f(x_1) &\stackrel{f(h_1)}{\to}& f(x_2) \\ \downarrow^{\mathrlap{\alpha}} && \downarrow^{\mathrlap{id}} \\ f(x_2) &\stackrel{f(h_2)}{\to}& f(x_2) } \,.

Therefore $h_2^{-1}\circ h_1$ is a preimage of $\alpha$ under $f$, and hence $f$ is full.

See also (eso+full, faithful) factorization system.

Related concepts

• n-truncated object in an (infinity,1)-category

• Eilenberg subcomplex

References

Section 6.5.1 of

• Jacob Lurie, Higher Topos Theory .

A discussion in terms of model category presentations is in section 8 of

• Charles Rezk, Toposes and homotopy toposes (pdf)