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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

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

See also

For related references see

Revised on June 5, 2014 12:04:27 by Urs Schreiber (92.68.97.89)