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

Idea

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

• locally connected space

• locally simply connected space

• locally $2$-connected space

• locally $n$-connected space

• locally contractible space

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

• locally connected topos

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

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

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

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

Definitions

Definition

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

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

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

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

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

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

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

Remark

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

Definition

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

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

Similarly we have:

Definition

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

For $n = 0$ 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 $C$ be a locally ∞-connected (∞,1)-site/∞-connected (∞,1)-site. Then the (∞,1)-category of (∞,1)-sheaves $Sh_{(\infty,1)}(C)$ is a locally $\infty$-connected $(\infty,1)$-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 $C$ has contractible objects, then there exists a left adjoint $Ho(\Pi):Ho(Sh_{(\infty,1)}(C)) \to Ho(\infty Grpd)$.

This includes the following examples.

Example

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

Over locally $n$-connected topological spaces

Example

For $X$ a locally contractible space, such that $Sh_{(\infty,1)}(X)$ is hypercomplete, $Sh_{(\infty,1)}(X)$ is a locally $\infty$-connected $(\infty,1)$-topos.

Proof

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

Proposition

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

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

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

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

$$\Pi(X) \simeq Sing X \,.$$

Proof

By using the presentations of $Sh_{(\infty,1)}(X)$ 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-$(\infty,1)$-toposes

Proposition

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

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

$$\Pi_X(X \in \mathbf{H}/X) \simeq \Pi(X \in \mathbf{H}) \,.$$

Proof

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

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

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

Remark

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

Proof

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

$$\Pi_X (X \to X) \simeq \Pi X_! (X \to X) \simeq \Pi X \simeq * \,.$$

Properties

Relation to slicing

Proposition

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

• the canonical morphism $\coprod_i U_i \to *$ out of their coproduct to the terminal object is an effective epimorphism;

• all the slice-(∞,1)-toposes $\mathcal{X}_{/U_i}$ are locally $\infty$-connected.

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

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

Relation to locally connected toposes

Proposition

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

Proof

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

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

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

$$\Pi_0 \coloneqq \tau_{\leq 0} \circ \Pi \,.$$

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

• Whitehead tower in an (∞,1)-topos.

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

Related concepts

• locally connected topos / locally ∞-connected (∞,1)-topos

  • connected topos / ∞-connected (∞,1)-topos

  • strongly connected topos / strongly ∞-connected (∞,1)-topos

  • totally connected topos / totally ∞-connected (∞,1)-topos

• local topos / local (∞,1)-topos.

• cohesive topos / cohesive (∞,1)-topos

and

• locally connected site, locally ∞-connected site

• connected site

• local site

• cohesive site, (∞,1)-cohesive site

References

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

• Carlos Simpson, Constantin Teleman, de Rham’s theorem for $\infty$-stacks (pdf)

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

• Jacob Lurie, Higher Algebra

See also

• Marc Hoyois, A note on Étale homotopy, 2013 (pdf)

For related references see

• geometric homotopy groups in an (∞,1)-topos

• cohesive (∞,1)-topos.