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shape of an (infinity,1)-topos

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

If an (∞,1)-topos H\mathbf{H} is that of (∞,1)-sheaves on (the site of open subsets of) a paracompact topological spaceH=Sh (,1)(X)\mathbf{H} = Sh_{(\infty,1)}(X) – then its shape is the strong shape of XX in the sense of shape theory: a pro-object Shape(X)Shape(X) in the category of CW-complexes.

It turns out that Shape(X)Shape(X) may be extracted in a canonical fashion from just the (∞,1)-topos Sh (,1)(X)Sh_{(\infty,1)}(X), and in a way that makes sense for any (∞,1)-topos. This then gives a definition of shape of general (,1)(\infty,1)-toposes.

Definition

Definition

The composite (∞,1)-functor

Π:(,1)ToposYFunc((,1)Topos,Grpd) opLex(PSh(),Grpd)AccLex(Grpd,Grpd) opProGrpd \Pi : (\infty,1)Topos \stackrel{Y}{\to} Func((\infty,1)Topos, \infty Grpd)^{op} \stackrel{Lex(PSh(-), \infty Grpd)}{\to} AccLex(\infty Grpd, \infty Grpd)^{op} \simeq Pro \infty Grpd

is the shape functor . Its value

Π(H)=(,1)Topos(H,PSh()) \Pi(\mathbf{H}) = (\infty,1)Topos(\mathbf{H}, PSh(-))

on an (,1)(\infty,1)-topos H\mathbf{H} is the shape of H\mathbf{H}.

Here

That this does indeed land in accessible left exact functors is shown below.

Properties

Notice that for every (∞,1)-topos H\mathbf{H} there is a unique geometric morphism

(LConstΓ):HΓLConstGrpd (LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd

where ∞Grpd is the (,1)(\infty,1)-topos of ∞-groupoids, Γ\Gamma is the global sections (∞,1)-functor and LConstLConst is the constant ∞-stack functor.

Proposition

The shape of H\mathbf{H} is the composite functor

Π(H):=ΓLConst:GrpdLConstHΓGrpd \Pi(\mathbf{H}) := \Gamma \circ LConst \;\;:\;\; \infty Grpd \stackrel{LConst}{\to} \mathbf{H} \stackrel{\;\;\Gamma\;\;}{\to} \infty Grpd

regarded as an object

Π(H)Pro(Grpd)=Lex(Grpd,Grpd) op. \Pi(\mathbf{H}) \in Pro(\infty Grpd) = Lex(\infty Grpd, \infty Grpd)^{op} \,.
Proof

For XX \in ∞Grpd we have by the (∞,1)-Grothendieck construction-theorem and using that up to equivalence every morphism of \infty-groupoids is a Cartesian fibration (see there) that

Func(X,Grpd)Grpd/X Func(X,\infty Grpd) \simeq \infty Grpd/X

is the over-(∞,1)-category. Moreover, by the theorem about limits in ∞Grpd we have that the terminal geometric morphism Hom(*,):[X,Grpd]GrpdHom(*,-): [X, \infty Grpd] \to \infty Grpd is the canonical projection Grpd/XGrpd\infty Grpd/ X \to \infty Grpd. This means that it is an etale geometric morphism. So for any geometric morphism f:H[X,Grpd]f : \mathbf{H} \to [X, \infty Grpd] we have a system of adjoint (∞,1)-functors

(LConstΓ):Hf *f *Grpd/Xπ *π *Grpd. (LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \infty Grpd/X \stackrel{\overset{\pi^*}{\leftarrow}}{\underset{\pi_*}{\to}} \infty Grpd \,.

whose composite is the global section geometric morphism as indicated, because that is terminal.

Notice that in Grpd/X\infty Grpd/X there is a canonical morphism

(*π *X):=(X(Id,Id)X×X). (* \to \pi^* X) := (X \stackrel{(Id,Id)}{\to} X \times X) \,.

The image of this under f *f^* is (using that this preserves the terminal object) a morphism

*f *π *X=LConstX * \to f^* \pi^* X = LConst X

in H\mathbf{H}.

Conversely, given a morphism of the form *LConstX* \to LConst X in H\mathbf{H} we obtain the base change geometric morphism

HH/*H/LConstXΓGrpd/X. \mathbf{H} \simeq \mathbf{H}/* \to \mathbf{H}/LConst X \stackrel{\Gamma}{\to} \infty Grpd/X \,.

One checks that these constructions establish an equivalence

(,1)Topos(H,Grpd/X)H(*,LConstX). (\infty,1)Topos(\mathbf{H}, \infty Grpd/X) \simeq \mathbf{H}(*, LConst X) \,.

Using this, we see that

Π(H):X (,1)Topos(H,X) H(*,LConstX) H(LConst*,LConstX) Grpd(*,ΓLConstX) ΓLConstX. \begin{aligned} \Pi (\mathbf{H}) : X \mapsto & (\infty,1)Topos(\mathbf{H}, X) \\ & \simeq \mathbf{H}(*,LConst X) \\ & \simeq \mathbf{H}(LConst *, LConst X) \\ & \simeq \infty Grpd(*, \Gamma LConst X) \\ & \simeq \Gamma LConst X \end{aligned} \,.
Remark

In particular this does show that Π(H):GrpdGrpd\Pi(\mathbf{H}) : \infty Grpd \to \infty Grpd does preserve finite (,1)(\infty,1)-limits, since Γ\Gamma preserves all limits and LConstLConst is a left exact functor. It also shows that it is accessible, since Γ\Gamma and LConstLConst are both accessible.

Examples

Shape of a locally \infty-connected topos

Suppose that H\mathbf{H} is locally ∞-connected, meaning that Γ\Gamma has a left adjoint Π\Pi which constructs the homotopy ∞-groupoids of objects of H\mathbf{H}. Then Shape(H)\Shape(\mathbf{H}) is represented by Π(*)Grpd\Pi(*)\in \infty Grpd, for we have

Shape(H)(A) =Γ(LConst(A)) =Hom Grpd(*,Γ(LConst(A))) =Hom H(LConst(*),LConst(A)) =Hom H(*,LConst(A)) =Hom Grpd(Π(*),A). \begin{aligned} Shape(\mathbf{H})(A) &= \Gamma(LConst(A))\\ &= Hom_{\infty Grpd}(*, \Gamma(LConst(A)))\\ &= Hom_{\mathbf{H}}(LConst(*), LConst(A)) \\ &= Hom_{\mathbf{H}}(*, LConst(A)) \\ &= Hom_{\infty Grpd}(\Pi(*),A). \end{aligned}

Thus, if we regard Π(*)\Pi(*) as “the fundamental ∞-groupoid of H\mathbf{H}” — which is reasonable since when H=Sh(X)\mathbf{H}=Sh(X) consists of sheaves on a locally contractible topological space XX, Π H(*)\Pi_{\mathbf{H}}(*) is equivalent to the usual fundamental ∞-groupoid of XX — then we can regard the shape of an (,1)(\infty,1)-topos as a generalized version of the “homotopy \infty-groupoid” which nevertheless makes sense even for non-locally-contractible toposes, by taking values in the larger category of “pro-\infty-groupoids.”

It follows also that H\mathbf{H} is not only locally ∞-connected but also ∞-connected, then it has the shape of a point.

Shape of a topological space

For a discussion of how the (,1)(\infty,1)-topos theoretic shape of Sh (,1)(X)Sh_{(\infty,1)}(X) relates to the ordinary shape-theoretic strong shape of the topological space XX see shape theory.

Shape of an essential retract

The following is trivial to observe, but may be useful to note.

Observation

Let (f !f *f *):Hf *f *f !B(f_! \dashv f^* \dashv f_*) : \mathbf{H} \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} \mathbf{B} be an essential geometric morphism of (,1)(\infty,1)-toposes that exhibits B\mathbf{B} as an essential retract of H\mathbf{H} in that

(IdId)Bf *f !Hf *f *B. (Id \dashv Id) \;\; \simeq \;\; \mathbf{B} \stackrel{\overset{f_!}{\leftarrow}}{\underset{f^*}{\to}} \mathbf{H} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathbf{B} \,.

Then the shape of B\mathbf{B} is equivalent to that of H\mathbf{H}.

Proof

Since Grpd\infty Grpd is the terminal object in the category of Grothendieck (,1)(\infty,1)-toposes and geometric morphisms, we have

(GrpdLConst BBΓ BGrpd) (GrpdLConst BBf *Hf *BΓ BGrpd) (GrpdLConst HHΓ HGrpd). \begin{aligned} (\infty Grpd \stackrel{LConst_{\mathbf{B}}}{\to} \mathbf{B} \stackrel{\Gamma_\mathbf{B}}{\to} \infty Grpd) &\simeq (\infty Grpd \stackrel{LConst_{\mathbf{B}}}{\to} \mathbf{B} \stackrel{f^*}{\to} \mathbf{H} \stackrel{f_*}{\to} \mathbf{B} \stackrel{\Gamma_\mathbf{B}}{\to} \infty Grpd) \\ &\simeq (\infty Grpd \stackrel{LConst_\mathbf{H}}{\to} \mathbf{H} \stackrel{\Gamma_\mathbf{H}}{\to} \infty Grpd) \end{aligned} \,.
Example

Every

over Grpd\infty Grpd has the shape of the point.

Proof

By definition H\mathbf{H} is \infty-connected if the constant ∞-stack inverse image f *=LConstf^* = L Const is

  1. not only a left but also a right adjoint;

  2. is a full and faithful (∞,1)-functor.

By standard properties of adjoint (∞,1)-functors we have that a right adjoint f *f^* is a full and faithful (∞,1)-functor precisely if the counit f !f *Idf_! f^* \to Id is an equivalence.

Equivalently, we can observe that a locally ∞-connected (∞,1)-topos is ∞-connected precisely when Π\Pi preserves the terminal object, and apply the above observation that the shape of a locally ∞-connected (∞,1)-topos is represented by Π(*)\Pi(*).

References

The definition of shape of (,1)(\infty,1)-toposes as ΓLConst\Gamma \circ LConst is due to

This and the relation to shape theory, more precisely the strong shape, of topological spaces is further discussed in section 7.1.6 of

Revised on February 11, 2011 21:44:25 by Zoran Škoda (161.53.130.104)