nLab
infinity-connected (infinity,1)-topos

Contents

Idea

If we think of an (∞,1)-topos as a generalized topological space, then it being ∞-connected is the analogue of a topological space being (weakly) contractible, i.e. weak-homotopy equivalent to a point.

It is an (∞,1)-categorification of the notion of a topos being connected.

Definition

Let H\mathbf{H} be a ((∞,1)-sheaf-)(,1)(\infty,1)-topos. It therefore admits a unique geometric morphism (LConstΓ):HΓ(L\Const\dashv\Gamma)\colon \mathbf{H}\xrightarrow{\Gamma} ∞Grpd given by global sections. We say that H\mathbf{H} is \infty-connected if LConstLConst is fully faithful.

More generally, we call a geometric morphism between (,1)(\infty,1)-toposes connected if its inverse image functor is fully faithful.

Properties

Observation

An \infty-connected (,1)(\infty,1)-topos has the shape of the point, in the sense of shape of an (∞,1)-topos.

Proof

By a basic property of adjoint (∞,1)-functors, LConstLConst being a full and faithful (∞,1)-functor is equivalent to the unit of (LConstΓ)(LConst \dashv \Gamma) being an equivalence

Id GrpdΓLConst. Id_{\infty Grpd} \stackrel{\simeq}{\to} \Gamma LConst \,.

By definition of shape of an (∞,1)-topos this means that H\mathbf{H} has the same shape as ∞Grpd, which is to say that it shape is represented, as a functor GrpdGrpd\infty Grpd \to \infty Grpd, by the terminal object **. Hence it has the “shape of the point”.

Locally ∞-connected and ∞-connected

As in the case of connected 1-topoi, we have the following.

Proposition

If an (,1)(\infty,1)-topos H\mathbf{H} is locally ∞-connected (i.e. LConstLConst has a left adjoint Π\Pi), then H\mathbf{H} is connected if and only if Π\Pi preserves the terminal object.

Proof

This is just like the 1-categorical proof. On the one hand, if H\mathbf{H} is ∞-connected, so that LConstLConst is fully faithful, then by properties of adjoint (∞,1)-functors this implies that the counit ΠLConstId\Pi \circ LConst \to \Id is an equivalence. But LConstLConst preserves the terminal object, since it is left exact, so Π(*)Π(LConst(*))*\Pi(*) \simeq \Pi(LConst(*)) \simeq *.

Conversely, suppose Π(*)*\Pi(*)\simeq *. Then any \infty-groupoid AA can be written as A=colim A*A = \colim^A *, the (∞,1)-colimit over AA itself of the constant diagram at the terminal object (see the details here). Since LConstLConst and Π\Pi are both left adjoints, both preserve colimits, so we have

Π(LConst(A))Π(LConst(colim A*))colim AΠ(LConst(*))colim A*A. \Pi(LConst(A)) \simeq \Pi(LConst(\colim^A *)) \simeq \colim^A \Pi(LConst(*)) \simeq \colim^A * \simeq A.

Therefore, the counit ΠLConstId\Pi \circ LConst \to \Id is an equivalence, so LConstLConst is fully faithful, and H\mathbf{H} is ∞-connected.

Revised on April 25, 2013 15:19:36 by Urs Schreiber (82.169.65.155)