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

$$(\Delta ⊣\Gamma ):H\stackrel{\Gamma }{\to }\mathrm{\infty }Grpd$$(\Delta\dashv\Gamma): \mathbf{H} \xrightarrow{\Gamma}\infty\Grpd

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

$$(\Pi ⊣\Delta ⊣\Gamma ):H\stackrel{\stackrel{\Pi }{\to }}{\stackrel{\stackrel{\Delta }{\leftarrow }}{\underset{\Gamma }{\to }}}\mathrm{\infty }\mathrm{Grpd}.$$(\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 $H$ is an ∞-connected (∞,1)-topos.

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

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

Remark

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

Definition

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

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

Similarly we have:

Definition

For $n\in \mathbb{N}$ an $(n+1,1)$-topos $H$ 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 $\mathrm{\infty }$-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 ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(C)$ is a locally $\mathrm{\infty }$-connected $(\mathrm{\infty },1)$-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 $\mathrm{Ho}(\Pi ):\mathrm{Ho}({\mathrm{Sh}}_{(\mathrm{\infty },1)}(C))\to \mathrm{Ho}(\mathrm{\infty }\mathrm{Grpd})$.

This includes the following examples.

Example

The sites CartSp${}_{\mathrm{top}}$ ${\mathrm{CartSp}}_{\mathrm{smooth}}$ ${\mathrm{CartSp}}_{\mathrm{synthdiff}}$ are locally $\mathrm{\infty }$-connected. The corresponding $(\mathrm{\infty },1)$-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, ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(X)$ is a locally $\mathrm{\infty }$-connected $(\mathrm{\infty },1)$-topos.

Proof

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

By the same kind of argument:

Example

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

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 ${\mathrm{Sh}}_{(inffty,1)}(C)$ under the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos-functor

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

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

$$\Pi (X)\simeq \mathrm{Sing}X.$$\Pi(X) \simeq Sing X \,.

Proof

By using the presentations of ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(X)$ 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 $\mathrm{\infty }$-connected over-$(\mathrm{\infty },1)$-toposes

Proposition

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

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

$${\Pi }_X(X\in H/X)\simeq \Pi (X\in H).$$\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

$$({\Pi }_X⊣{\Delta }_X⊣{\Gamma }_X):H_{/X}\stackrel{\stackrel{X_{!}}{\to }}{\stackrel{\stackrel{X^{*}}{\leftarrow }}{\underset{X_{*}}{\to }}}H\stackrel{\stackrel{\Pi }{\to }}{\stackrel{\stackrel{\Delta }{\leftarrow }}{\underset{\Gamma }{\to }}}\mathrm{\infty }\mathrm{Grpd}$$(\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 $(Y\to X)\in H/X$ to $Y\in H$.

Remark

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

Proof

By the discussion there we need to check that ${\Pi }_X$ preserves the terminal object:

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

Properties

Proposition

Let $𝒳$ be an $(\mathrm{\infty },1)$-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 over-(∞,1)-toposes ${𝒳}_{/U_i}$ are locally $\mathrm{\infty }$-connected.

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

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

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 $\mathrm{\infty }$-connected $(\mathrm{\infty },1)$-toposes is around proposition 2.18 of

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

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

• Jacob Lurie, Higher Algebra

For related references see

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

• cohesive (∞,1)-topos.