nLab (∞,1)-local geometric morphism

Idea

The analog in the context of (∞,1)-topos theory of a local geometric morphism in topos theory.

Definition

Definition

A local (∞,1)-geometric morphism $f : \mathbf{H} \to \mathbf{S}$ between (∞,1)-toposes $\mathbf{H},\mathbf{S}$ is

• an (∞,1)-geometric morphism

  $$(f^* \dashv f_*) : \mathbf{H} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*} {\to}} \mathbf{S}$$

• such that

  1. a further right adjoint $f^! : \mathbf{S} \to \mathbf{H}$ to the direct image functor exists:

     $$(f^* \dashv f_* \dashv f^!) : \mathbf{H} \stackrel{\overset{f^*}{\leftarrow}} {\stackrel{\underset{f_*}{\to}}{\underset{f^!}{\leftarrow}}} \mathbf{S}$$

  2. and $f$ is a ∞-connected (∞,1)-geometric morphism.

If $f : \mathbf{H} \to \mathbf{S}$ is the global section (∞,1)-geometric morphism in the over-(∞,1)-category Topos$/\mathbf{S}$, then we say that $\mathbf{H}$ is a local $(\infty,1)$-topos over $\mathbf{S}$.

Remark

If $\mathbf{S} =$ ∞Grpd then the extra condition that $f$ is ∞-connected (∞,1)-geometric morphism is automatic (see Properties – over ∞Grpd).

Properties

Over $\infty Grpd$

Proposition

Every local $(\infty,1)$-topos over ∞Grpd has homotopy dimension $\leq 0$.

See homotopy dimension for details.

Proposition

If an (∞,1)-geometric morphism $f : \mathbf{H} \to$ ∞Grpd has an extra right adjoint $f^!$ to its direct image, then $\mathbf{H}$ is an ∞-connected (∞,1)-topos.

Proof

By the general properties of adjoint (∞,1)-functors it is sufficient to show that $f_! f^* \simeq Id$. To see this, we use that every ∞-groupoid $S \in$ ∞Grpd is the (∞,1)-colimit (as discussed there) over itself of the (∞,1)-functor constant on the point: $S \simeq {\lim_\to}_{S} *$.

The left adjoint $f^*$ preserves all (∞,1)-colimits, but if $f_*$ has a right adjoint, then it does, too, so that for all $S$ we have

$$f_* f^* {\lim_\to}_S * \simeq {\lim_\to}_S f_* f^* * \,.$$

Now $f_*$, being a right adjoint preserves the terminal object and so does $f^*$ by definition of (∞,1)-geometric morphism. Therefore

$$\cdots \simeq {\lim_\to}_S * \simeq S \,.$$

Concrete objects

Every local $(\infty,1)$-geometric morphism induces a notion of concrete (∞,1)-sheaves. See there for more (also see cohesive (∞,1)-topos).

Examples

Local over-$(\infty,1)$-toposes

Proposition

Let $\mathbf{H}$ be any (∞,1)-topos (over ∞Grpd) and let $X \in \mathbf{H}$ be an object that is small-projective. Then the over-(∞,1)-topos $\mathbf{H}/X$ is local.

Proof

We check that the global section (∞,1)-geometric morphism $\Gamma : \mathbf{H}/X \to$ ∞Grpd preserves (∞,1)-colimits.

The functor $\Gamma$ is given by the hom-functor out of the terminal object of $\mathcal{H}/X$, this is $(X \stackrel{Id}{\to} X)$:

$$\Gamma : (A \stackrel{f}{\to} X) \mapsto Hom_{\mathbf{H}/X}(Id_X, f) \,.$$

The hom-∞-groupoids in the over-(∞,1)-category are (as discussed there) homotopy fibers of the hom-sapces in $\mathbf{H}$: we have an (∞,1)-pullback diagram

$$\array{ \mathbf{H}/X(Id_X, (A \to X)) &\to& \mathbf{H}(X,A) \\ \downarrow && \downarrow^{f_*} \\ * &\stackrel{Id_X}{\to}& \mathbf{H}(X,X) } \,.$$

Overserve that (∞,1)-colimits in the over-(∞,1)-category $\mathbf{H}/X$ are computed in $\mathbf{H}/X$.

$${\lim_{\to}}_i (A_i \stackrel{f_i}{\to} X) \simeq ({\lim_\to}_i A_i) \to X \,.$$

If $X$ is small-projective then by definition we have

$$\mathbf{H}(X, {{\lim}_\to}_i A_i) \simeq {\lim_\to}_i \mathbf{H}(X, A_i) \,,$$

Inserting all this into the above $(\infty,1)$-pullback gives the $(\infty,1)$-pullback

$$\array{ \mathbf{H}/X(Id_X, {\lim_\to}_i (A_i \to X)) &\to& {\lim_\to}_i \mathbf{H}(X, A_i) \\ \downarrow && \downarrow^{f_*} \\ * &\stackrel{Id_X}{\to}& \mathbf{H}(X,X) } \,.$$

By universal colimits in the (∞,1)-topos ∞Grpd, this (∞,1)-pullback of an (∞,1)-colimit is the $(\infty,1)$-colimit of the separate pullbacks, so that

$$\Gamma({\lim_\to}_i (A_i \to X))) \simeq \mathbf{H}/X(Id_X, {\lim_\to}_i (A_i \to X)) \simeq {\lim_\to}_i \mathbf{H}/X(Id_X,(A_i \to X)) \simeq {\lim_\to}_i \Gamma(A_i \to X) \,.$$

So $\Gamma$ does commute with colimits if $X$ is small-projective. Since all (∞,1)-toposes are locally presentable (∞,1)-categories it follows by the adjoint (∞,1)-functor that $\Gamma$ has a right adjoint (∞,1)-functor.

Pyknotic $\infty$-groupoids

The $(\infty,1)$-topos of pyknotic $\infty$-groupoids is a local $(\infty,1)$-topos, according to Zhu 2023.

Related concpepts

• 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

The 1-categorical notion is discussed in

• Peter Johnstone, Ieke Moerdijk, Local maps of toposes Proc. London Math. Soc. (1989) s3-58 (2): 281-305. (pdf)

That pyknotic $\infty$-groupoids (note the author uses “condensed” to mean “pyknotic”) form a local $(\infty,1)$-topos:

• Qi Zhu, Fractured Structure on Condensed Anima, MSc thesis (2023) [pdf, pdf]