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(∞,1)-local geometric morphism

Context

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

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

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:HSf : \mathbf{H} \to \mathbf{S} between (∞,1)-toposes H,S\mathbf{H},\mathbf{S} is

  • an (∞,1)-geometric morphism

    (f *f *):Hf *f *S (f^* \dashv f_*) : \mathbf{H} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*} {\to}} \mathbf{S}
  • such that

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

      (f *f *f !):Hf !f *f *S (f^* \dashv f_* \dashv f^!) : \mathbf{H} \stackrel{\overset{f^*}{\leftarrow}} {\stackrel{\underset{f_*}{\to}}{\underset{f^!}{\leftarrow}}} \mathbf{S}
    2. and ff is a ∞-connected (∞,1)-geometric morphism.

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

Remark

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

Properties

Over Grpd\infty Grpd

Proposition

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

See homotopy dimension for details.

Proposition

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

Proof

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

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

f *f *lim S*lim Sf *f **. f_* f^* {\lim_\to}_S * \simeq {\lim_\to}_S f_* f^* * \,.

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

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

Concrete objects

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

Examples

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

Proposition

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

Proof

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

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

Γ:(AfX)Hom H/X(Id X,f). \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 H\mathbf{H}: we have an (∞,1)-pullback diagram

H/X(Id X,(AX)) H(X,A) f * * Id X H(X,X). \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 H/X\mathbf{H}/X are computed in H/X\mathbf{H}/X.

lim i(A if iX)(lim iA i)X. {\lim_{\to}}_i (A_i \stackrel{f_i}{\to} X) \simeq ({\lim_\to}_i A_i) \to X \,.

If XX is small-projective then by definition we have

H(X,lim iA i)lim iH(X,A i), \mathbf{H}(X, {{\lim}_\to}_i A_i) \simeq {\lim_\to}_i \mathbf{H}(X, A_i) \,,

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

H/X(Id X,lim i(A iX)) lim iH(X,A i) f * * Id X H(X,X). \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 (,1)(\infty,1)-colimit of the separate pullbacks, so that

Γ(lim i(A iX)))H/X(Id X,lim i(A iX))lim iH/X(Id X,(A iX))lim iΓ(A iX). \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 Γ\Gamma does commute with colimits if XX is small-projective. Since all (∞,1)-toposes are locally presentable (∞,1)-categories it follows by the adjoint (∞,1)-functor that Γ\Gamma has a right adjoint (∞,1)-functor.

and

References

The 1-categorical notion is discussed in

Revised on November 19, 2013 11:49:28 by Urs Schreiber (77.251.114.72)