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locally n-connected (n+1,1)-topos

Context

(,1)(\infty,1)-Topos theory

Contents

Idea

A topos may be thought of as a generalized topological space. Accordingly, the notions of

have analogs for toposes, (n,1)-toposes and (∞,1)-toposes

  • locally connected topos

  • locally simply connected (2,1)-topos?

  • locally nn-connected (n+1,1)(n+1,1)-topos

  • locally \infty-connected (,1)(\infty,1)-topos

The numbering mismatch is traditional from topology; see n-connected space. It reads a bit better if we say locally nn-simply connected for locally nn-connected, since locally 11-(simply) connected is locally simply connected, but being locally nn-simply connected is still a property of an (n+1,1)(n+1,1)-topos.

Definitions

Definition

A (∞,1)-sheaf (∞,1)-topos H\mathbf{H} is called locally \infty-connected if the (essentially unique) global section (∞,1)-geometric morphism

(ΔΓ):HΓGrpd (\Delta\dashv\Gamma): \mathbf{H} \xrightarrow{\Gamma}\infty\Grpd

extends to an essential geometric morphism (,1)(\infty,1)-geometric morphism, i.e. there is a further left adjoint Π\Pi

(ΠΔΓ):HΓΔΠGrpd. (\Pi \dashv \Delta \dashv \Gamma) : \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd \,.

If in addition Π\Pi preserves the terminal object we say that H\mathbf{H} is an ∞-connected (∞,1)-topos.

If Π\Pi preserves even all finite (∞,1)-products we say that H\mathbf{H} is a strongly ∞-connected (∞,1)-topos.

If Π\Pi preserves even all finite (∞,1)-limits we say that H\mathbf{H} is a totally ∞-connected (∞,1)-topos.

Remark

In (Lurie, section A.1) this is called an (,1)(\infty,1)-topos of locally constant shape.

Definition

For H\mathbf{H} a locally \infty-connected (,1)(\infty,1)-topos and XHX \in \mathbf{H} an object, we call ΠX\Pi X \in ∞Grpd the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos of XX. The (categorical) homotopy groups of Π(X)\Pi(X) we call the geometric homotopy groups of XX

π geom(X):=π (Π(X)). \pi_\bullet^{geom}(X) := \pi_\bullet(\Pi (X)) \,.

Similarly we have:

Definition

For nn \in \mathbb{N} an (n+1,1)(n+1,1)-topos H\mathbf{H} is called locally nn-connected if the (essentially unique) global section geometric morphism is has an extra left adjoint.

For n=0n = 0 this reproduces the case of a locally connected topos.

Examples

Over locally \infty-connected sites

The follow proposition gives a large supply of examples.

Proposition

Let CC be a locally ∞-connected (∞,1)-site/∞-connected (∞,1)-site. Then the (∞,1)-category of (∞,1)-sheaves Sh (,1)(C)Sh_{(\infty,1)}(C) is a locally \infty-connected (,1)(\infty,1)-topos.

See locally ∞-connected (∞,1)-site/∞-connected (∞,1)-site for the proof.

Remark

In (SimpsonTeleman, prop. 2.18) is stated essentially what the above proposition asserts at the level of homotopy categories: if CC has contractible objects, then there exists a left adjoint Ho(Π):Ho(Sh (,1)(C))Ho(Grpd)Ho(\Pi):Ho(Sh_{(\infty,1)}(C)) \to Ho(\infty Grpd).

This includes the following examples.

Example

The sites CartSp top{}_{top} CartSp smoothCartSp_{smooth} CartSp synthdiffCartSp_{synthdiff} are locally \infty-connected. The corresponding (,1)(\infty,1)-toposes are the cohesive (∞,1)-toposes ETop∞Grpd, Smooth∞Grpd and SynthDiff∞Grpd.

Over locally nn-connected topological spaces

Example

For XX a locally contractible space, Sh (,1)(X)Sh_{(\infty,1)}(X) is a locally \infty-connected (,1)(\infty,1)-topos.

Proof

The full subcategory cOp(X)Op(X)cOp(X) \hookrightarrow Op(X) of the category of open subsets on the contractible subsets is another site of definition for Sh (,1)(X)Sh_{(\infty,1)}(X). And it is a locally ∞-connected (∞,1)-site.

By the same kind of argument:

Example

For nn \in \mathbb{N} and for XX a locally nn-connected topological space, Sh (n+1,1)(X)Sh_{(n+1,1)}(X) is a locally nn-connected (n+1)(n+1)-topos.

Proposition

For XX a locally contractible topological space we have that the fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos computes the correct homotopy type of XX:

the image of XX as the terminal object in Sh (inffty,1)(C)Sh_{(\inffty,1)}(C) under the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos-functor

Π:Sh (,1)(X)Grpd \Pi : Sh_{(\infty,1)}(X) \to \infty Grpd

is equivalent to the ordinary fundamental ∞-groupoid given by the singular simplicial complex

Π(X)SingX. \Pi(X) \simeq Sing X \,.
Proof

By using the presentations of Sh (,1)(X)Sh_{(\infty,1)}(X) by the model structure on simplicial presheaves as discussed at locally ∞-connected (∞,1)-site one finds that this boils down to the old Artin-Mazur theorem. More on this at geometric homotopy groups in an (∞,1)-topos.

Locally \infty-connected over-(,1)(\infty,1)-toposes

Proposition

For H\mathbf{H} a locally \infty-connected (,1)(\infty,1)-topos, also all its objects XHX \in \mathbf{H} are locally \infty-connected, in that their petit over-(∞,1)-toposes H/X\mathbf{H}/X are locally \infty-connected.

The two notions of fundamental \infty-groupoids of XX induced this way do agree, in that there is a natural equivalence

Π X(XH/X)Π(XH). \Pi_X(X \in \mathbf{H}/X) \simeq \Pi(X \in \mathbf{H}) \,.
Proof

By the general facts recalled at etale geometric morphism we have a composite essential geometric morphism

(Π XΔ XΓ X):H /XX *X *X !HΓΔΠGrpd (\Pi_X \dashv \Delta_X \dashv \Gamma_X) : \mathbf{H}_{/X} \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{\X_*}{\to}}} \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd

and X !X_! is given by sending (YX)H/X(Y \to X) \in \mathbf{H}/X to YHY \in \mathbf{H}.

Remark

If in the above XX is contractible in that ΠX*\Pi X \simeq * then H/X\mathbf{H}/X is even an ∞-connected (∞,1)-topos.

Proof

By the discussion there we need to check that Π X\Pi_X preserves the terminal object:

Π X(XX)ΠX !(XX)ΠX*. \Pi_X (X \to X) \simeq \Pi X_! (X \to X) \simeq \Pi X \simeq * \,.

Properties

Relation to slicing

Proposition

Let 𝒳\mathcal{X} be an (,1)(\infty,1)-topos and {U i} i\{U_i\}_i a collection of objects such that

Then also 𝒳\mathcal{X} itself is locally \infty-connected.

This appears as (Lurie, corollary A.1.7).

Relation to locally connected toposes

Proposition

For (ΠΔΓ):HGrpd(\Pi \dashv \Delta \dashv \Gamma) : \mathbf{H} \to \infty Grpd a locally \infty-connected (,1)(\infty,1)-topos, its underlying (1,1)-topos τ 0H\tau_{\leq 0} \mathbf{H} is a locally connected topos. Moreover, if H\mathbf{H} is strongly connected (the extra left adjoint preserves finite products), then so is τ 0H\tau_{\leq 0} \mathbf{H}.

Proof

The global sections geometric morphism ΓH(*,)\Gamma \simeq \mathbf{H}(*,-) is given by homming out of the terminal object and hence preserves 0-truncated objects by definition. Also, by the (ΠdahsvΔ)(\Pi \dahsv \Delta)-adjunction so does Δ\Delta: for every SSetτ GrpdGrpdS \in Set \simeq \tau_{\leq }\infty Grpd \hookrightarrow \infty Grpd and every XHX \in \mathbf{H} we have

H(X,Δ(S))Grpd(Π(X),S)Set(τ 0Π(X),S)SetGrpd. \mathbf{H}(X, \Delta(S)) \simeq \infty Grpd(\Pi(X), S) \simeq Set(\tau_{\leq 0} \Pi(X), S) \in Set \hookrightarrow \infty Grpd \,.

Therefore by essential uniqueness of adjoints the restriction Δ 0:SetGrpdΔH\Delta|_{\leq 0} \colon Set \hookrightarrow \infty Grpd \stackrel{\Delta}{\to} \mathbf{H} has a left adjoint given by

Π 0τ 0Δ. \Pi_0 \coloneqq \tau_{\leq 0} \circ \Delta \,.

Finally, by the discussion here, τ 0\tau_{\leq 0} preserves finite limits. Hence Π 0\Pi_0 does so if Π\Pi does.

Further structures

The fact that the terminal geometric morphism is essential gives rise to various induced structures of interest. For instance it induces a notion of

For a more exhaustive list of extra structures see cohesive (∞,1)-topos.

and

References

Some discussion of the homotopy category of locally \infty-connected (,1)(\infty,1)-toposes is around proposition 2.18 of

Undet the term locally constant shape the notion appears in section A.1 of

For related references see

Revised on November 8, 2012 17:56:29 by Urs Schreiber (82.169.65.155)