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 $\mathbf{H}$ is called $n$-connected for $-1 \leq n \in \mathbb{Z}$ if

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

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

   $$\pi_k(X) = * \;\;\; \text{for} \; k \leq n \,.$$

A morphism $f : X \to Y$ in an $(\infty,1)$-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 $\mathbf{H}_{/Y}$ 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, def. , precisely if its n-truncation $\tau_{\leq n} X$ is the terminal object of $\mathbf{H}$ (hence precisely if it is $\tau_{\leq n}$-comodal).

This is HTT, prop. 6.5.1.12.

Lemma

Every equivalence is $\infty$-connected.

This is HTT, prop. 6.5.1.16, item 2.

Remark

In a general $(\infty,1)$-topos the converse is not true: not every $\infty$-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

$$\Delta_f : X \to X \times_Y X$$

is $(n-1)$-connected.

This appears as HTT, prop. 6.5.1.18.

Factorization system

Proposition

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

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).

Blakers-Massey theorem

• Blakers-Massey theorem

The truncated / connected clock

In a hypercomplete (∞,1)-topos the $\infty$-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, $\cdots$;

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

• monomorphism = $(-1)$-truncated

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

• $\infty$-truncated = any morphism

Examples

In $Top$

In the the (∞,1)-category $L_{whe}$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 $\infty$-connected object is a contractible space.

More generally, a continuous function represents an $n$-connected morphism in $L_{whe} Top$ precisely if it is an n-connected continuous function (“n-equivalence”).

In $Grpd$

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. 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 $(\hat \alpha,id) : (x_1, x_2) \to (x_2, x_2)$ in $X \times_Y X$

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

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

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

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

• n-connective object

• Eilenberg subcomplex

• connective spectrum, connective cover

• Freudenthal suspension theorem

• Blakers-Massey theorem

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)