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

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

Idea

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

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

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

The Whitehead tower construction produces nn-connected objects.

Definition

Definition

An object XX in an (∞,1)-topos H\mathbf{H} is called nn-connected for 1n -1 \leq n \in \mathbb{Z} if

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

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

    π k(X)=*forkn. \pi_k(X) = * \;\;\; for k \leq n \,.

A morphism f:XYf : X \to Y in an (,1)(\infty,1)-topos is called nn-connected if

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

  2. regarded as an object in the over-(∞,1)-category H /Y\mathbf{H}_{/Y} all categorical homotopy groups equal to or below degree nn are trivial.

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

One adopts the following convenient terminology.

  • Every object is (2)(-2)-connected.

  • A (1)(-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)(-1)-connected morphisms. For more on this see n-connected/n-truncated factorization system.

Properties

General

Proposition

An object XX is nn-connected (for n2n \geq -2) precisely if its n-truncation τ nX\tau_{\leq n} X is the terminal object of H\mathbf{H}.

This is HTT, prop. 6.5.1.12.

Observation

Every equivalence is \infty-connected.

This is HTT, prop. 6.5.1.16, item 2.

Remark

In a general (,1)(\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 nn-connected morphisms is stable under pullback and pushout.

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

This is HTT, prop. 6.5.1.16, item 6.

Recursive characterization

Proposition

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

Δ f:XX× YX \Delta_f : X \to X \times_Y X

is (n1)(n-1)-connected.

This appears as HTT, prop. 6.5.1.18.

Factorization system

Proposition

Let H\mathbf{H} be an (∞,1)-topos. For all (2)n(-2) \leq n \leq \infty the class of nn-connected morphisms in H\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 H\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

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)(-2)-connected

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

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

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

  • monomorphism = (1)(-1)-truncated

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

  • \infty-truncated = any morphism

Examples

In TopTop

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

In GrpdGrpd

Proposition

Let f:XYf : X \to Y be a functor between groupoids. Regarded as a morphism in ∞Grpd ff 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 ff, being 0-connected is equivalent to being full.

The homotopy pullback X× YXX \times_Y X is given by the groupoid whose objects are triples (x 1,x 2X,α:f(x 1)f(x 2))(x_1, x_2 \in X, \alpha : f(x_1) \to f(x_2)) and whose morphisms are corresponding tuples of morphisms in XX making the evident square in YY commute.

By prop. 3 it is sufficient to check that the diagonal functor XX× YXX \to X \times_Y X is (-1)-connected, hence, as before, essentially surjective, precisely if ff is full.

First assume that ff is full. Then for (x 1,x 2,α)X× YX(x_1,x_2, \alpha) \in X \times_Y X any object, by fullness of ff there is a morphism α^:x 1x 2\hat \alpha : x_1 \to x_2 in XX, such that f(α^)=αf(\hat \alpha) = \alpha.

Accordingly we have a morphism (α^,id):(x 1,x 2)(x 2,x 2)(\hat \alpha,id) : (x_1, x_2) \to (x_2, x_2) in X× YXX \times_Y X

f(x 1) f(α^) f(x 2) α id f(x 2) id f(x 2) \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 2Xx_1, x_2 \in X such that there is a morphism α:f(x 1)f(x 2)\alpha : f(x_1) \to f(x_2) we are guaranteed morphisms h 1:x 1x 2h_1 : x_1 \to x_2 and h 2:x 2x 2h_2 : x_2 \to x_2 such that

f(x 1) f(h 1) f(x 2) α id f(x 2) f(h 2) f(x 2). \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 1h 1h_2^{-1}\circ h_1 is a preimage of α\alpha under ff, and hence ff is full.

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

References

Section 6.5.1 of

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

Revised on September 29, 2013 01:20:57 by Urs Schreiber (89.204.139.252)