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shape of an (infinity,1)-topos

Contents

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

To the extent that an (∞,1)-topos may be thought of as representing a space, the shape of the \infty-topos is the underlying pro-homotopy type of this space (Def. below). Equivalently, the shape of an \infty-topos is the generalized étale homotopy-type of its terminal object (Def. , Prop. below).

In the special case that the (∞,1)-topos is that of (∞,1)-sheaves on (the site of open subsets of) a paracompact topological space, its shape coincides with the strong shape of XX in the classical sense of shape theory.

In the special case that the \infty-topos is a slice of a cohesive ( , 1 ) (\infty,1) -topos H\mathbf{H} over some object, its shape is the cohesive shape of that object (Prop. below).

Definition

We state three somewhat different-looking definitions, and show that they are all equivalent to each other.

Definition

(shape of an \infty-topos – Toën-Vezzosi 2002, Def. 5.3.2)
The composite (∞,1)-functor

Shp:(,1)ToposYFunc((,1)Topos,Grpd) opLex(PSh(),Grpd)AccLex(Grpd,Grpd) opProGrpd Shp \;\colon\; (\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

Shp(H)=(,1)Topos(H,PSh()) Shp(\mathbf{H}) = (\infty,1)Topos \big( \mathbf{H} ,\, PSh(-) \big)

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 follows from the equivalence to the following definition, see Rem. below.

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

(LConstΓ):HΓLConstGrpd, (LConst \dashv \Gamma) \;\colon\; \mathbf{H} \underoverset { \underset{\Gamma}{\longrightarrow} } { \overset{LConst}{\longleftarrow} } { \bot } \infty Grpd \,,

where

Definition

(shape of an \infty-topos – Lurie 2009, Def. 7.1.6.3)
The shape of an ( , 1 ) (\infty,1) -topos H\mathbf{H} is the composite ( , 1 ) (\infty,1) -functor

Shp(H)ΓLConst:GrpdLConstHΓGrpd Shp(\mathbf{H}) \;\coloneqq\; \Gamma \circ LConst \;\;:\;\; \infty Grpd \stackrel{LConst}{\to} \mathbf{H} \stackrel{\;\;\Gamma\;\;}{\to} \infty Grpd

regarded as a pro- \infty -groupoid:

Shp(H)Pro(Grpd)=Lex(Grpd,Grpd) op. Shp(\mathbf{H}) \;\in\; Pro(\infty Grpd) \;=\; Lex(\infty Grpd, \infty Grpd)^{op} \,.

Proposition

Def. and Def. are equivalent.

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 slice (∞,1)-category. Moreover, by this 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) \;\colon\; \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) \;\coloneqq\; (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 in H\mathbf{H} of the form

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

Conversely, given a morphism of the form *LConstX* \to LConst X in H\mathbf{H} we obtain the base change geometric morphism of slice ( , 1 ) (\infty,1) -toposes

HH /*H /LConstXΓGrpd /X. \mathbf{H} \,\simeq\, \mathbf{H}_{/*} \xrightarrow{\;\;} \mathbf{H}_{/LConst X} \xrightarrow{\;\; \Gamma\;\;} \infty Grpd_{/X} \,.

One checks that these constructions establish an equivalence

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

Using this, we find the following sequence of equiavlences:

Shp(H):X (,1)Topos(H,Grpd /X) H(*,LConstX) H(LConst*,LConstX) Grpd(*,ΓLConstX) ΓLConst(X). \begin{aligned} Shp (\mathbf{H}) \;\colon\; X \;\; \mapsto \;\; & (\infty,1)Topos \big( \mathbf{H} ,\, \infty Grpd_{/X} \big) \\ & \simeq \mathbf{H}(*,LConst X) \\ & \simeq \mathbf{H}(LConst *, LConst X) \\ & \simeq \infty Grpd(*, \Gamma LConst X) \\ & \simeq \Gamma \circ LConst(X) \end{aligned} \,.

The composite of these is the equivalence to be shown.

Remark

Prop. immediately implies that Shp(H):GrpdGrpdShp(\mathbf{H}) \;\colon\; \infty Grpd \to \infty Grpd according to Def. does preserve finite (,1)(\infty,1)-limits, since in the equivalent Def. this is manifest: There Γ\Gamma clearly preserves all limits, since it is a right adjoint, and LConstLConst preserves finite limits, since it is a left exact functor by definition. Similarly, this makes manifest that Shp(H)Shp(\mathbf{H}) is accessible, since Γ\Gamma and LConstLConst are both accessible.

Definition

(shape of an \infty-topos – Hoyois 2013, p. 3)
For (ΓLConst):HGrpd(\Gamma \dashv LConst) \;\colon\; \mathbf{H} \to \infty Grpd an ( , 1 ) (\infty,1) -topos, write (with convenient overloading of notation)

(1)Shp: H Pro(Grpd) X H(X,LConst()) \array{ Shp \;\colon\; & \mathbf{H} &\longrightarrow& Pro(\infty Grpd) \\ & X &\mapsto& \mathbf{H} \big( X ,\, LConst(-) \big) }

for the pro-left adjoint to LConst:GrpdH,LConst \,\colon\, \infty Grpd \to \mathbf{H}, and say that the shape of H\mathbf{H} is the image under this pro-left adjoint of its terminal object * H\ast_{\mathbf{H}}:

Shp(H)Shp(* H). Shp(\mathbf{H}) \;\coloneqq\; Shp(\ast_{\mathbf{H}}) \,.

Proposition

Def. is equivalent to Def.

Proof

Consider the following sequence of natural equivalences:

Shp(H) ΓLConst() Grpd(*,ΓLConst()) H(LConst(*),LConst()) H(* H,LConst()) Shp(* H) \begin{aligned} Shp(\mathbf{H}) & \;\simeq\; \Gamma \circ LConst(-) \\ & \;\simeq\; \infty Grpd \big( \ast ,\, \Gamma \circ LConst(-) \big) \\ & \;\simeq\; \mathbf{H} \big( LConst(\ast) ,\, LConst(-) \big) \\ & \;\simeq\; \mathbf{H} \big( \ast_{\mathbf{H}} ,\, LConst(-) \big) \\ & \;\simeq\; Shp(\ast_{\mathbf{H}}) \end{aligned}

Here the line is just Def. (alternatively: is Prop. ), and the second line follows by the cartesian closure of ∞Grpd. The third line is the characteristic hom-equivalences of the adjunction ΓLConst\Gamma \dashv LConst . In the second but last step we use that LConstLConst, being a left exact functor, preserves terminal objects. The last step is the definition (1) of the pro-left adjoint.


Examples

Shape of locally \infty-connected toposes

Proposition

If H\mathbf{H} is locally ∞-connected, in that LConst\LConst has an actual left ( , 1 ) (\infty,1) -functor ShpShp (see at shape modality, in constrast to just a pro-left adjoint (1)) then the shape of H\mathbf{H} is Shp(*)GrpdPropGrpdShp(*) \in \infty Grpd \xhookrightarrow Prop \infty Grpd (under the embedding here):

Shp(H)Shp(*) Shp(\mathbf{H}) \;\; \simeq \;\; Shp(\ast)

Proof

This follows immediately from Prop. and the observation that an actual left adjoint, when it exists, of coincides with the pro-left adjoint under the embedding GrpdPro(Grpd)\infty Grpd \hookrightarrow Pro(\infty Grpd) (here)

But we may also see explicitly that we have the following sequence of natural equivalences of ∞-groupoids, starting with Def. (alternatively: Prop. (.

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

Thus, if we regard Shp(*)Shp(*) 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, Shp H(*)Shp_{\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.

More generally:

Remark

Let (f !f *f *):HB(f_! \dashv f^* \dashv f_*) : \mathbf{H} \xrightarrow{\;\;} \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} \underoverset {\underset{f^*}{\longrightarrow}} {\overset{f_!}{\longleftarrow}} {\;\;\;\bot\;\;\;} \mathbf{H} \underoverset {\underset{f_*}{\longrightarrow}} {\overset{f^*}{\longleftarrow}} {\;\;\;\bot\;\;\;} \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 ( , 1 ) (\infty,1) -category of Grothendieck ( , 1 ) (\infty,1) -toposes and geometric morphisms (see here), we have

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

Example

(cohesive (∞,1)-topos has trivial shape)
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 ShpShp preserves the terminal object, and apply the above observation that the shape of a locally ∞-connected (∞,1)-topos is represented by Shp(*)Shp(*).

Remark

(trivial shape of gros \infty-toposes)
That cohesive ( , 1 ) (\infty,1) -toposes have trivial shape (Exp. ) is a reflection of their characteristic nature as gros toposes: Rather than representing a single specific non-trivial space, cohesive \infty-toposes are gros \infty-categories of spaces (of geometric/cohesive spaces, specifically).

This is further brought out by Prop. below, in view of which Exp. , may be read as saying that the shape of a cohesive \infty-topos does not interfere with the cohesive shapes of its objects.

Shape of slice of a cohesive \infty-topos

Proposition

If H\mathbf{H} is a cohesive ( , 1 ) (\infty,1) -topos with shape modality

ʃ:HShpGrp DiscH ʃ \;\; \colon \;\; \mathbf{H} \xrightarrow{\;Shp\;} \Grp_\infty \xrightarrow{\;Disc\;} \mathbf{H}

then for every object XHX \,\in\, \mathbf{H} the shape of the slice ( , 1 ) (\infty,1) -topos H /X\mathbf{H}_{/X}, according to Def. , is equivalently the cohesive shape of XX:

Shp(H /X)Shp(X)Grp Pro(Grp ). Shp \big( \mathbf{H}_{/X} \big) \;\; \simeq \;\; Shp(X) \;\;\;\;\; \in \; Grp_\infty \xhookrightarrow{\;} Pro(Grp_\infty) \,.

Proof

For H\mathbf{H} any ( , 1 ) (\infty,1) -topos and XHX \,\in\, \mathbf{H} an object, the slice ( , 1 ) (\infty,1) -topos H /X\mathbf{H}_{/X} is related to H\mathbf{H} by the base change adjoint triple shown on the left here, together with, on the right, part of the adjoint quadruple that exhibits the cohesion of H\mathbf{H}:

By essential uniqueness of adjoint ( , 1 ) (\infty,1) -functors (here) and of the terminal ( , 1 ) (\infty,1) -geometric morphism (here), the composite adjunction is the global section geometric morphism of the slice topos:

(LConst XΓ X)(X×Disc()Pnts X). \big( \mathrm{LConst}_X \;\; \dashv \;\; \Gamma_X \big) \;\;\;\;\simeq\;\;\;\; \big( X \times \mathrm{Disc} (-) \;\; \dashv \;\; \mathrm{Pnts} \circ \prod_X \big) \,.

Hence, by Prop. , we need to exhibit a natural equivalence of this form:

(2)Γ XLConst X()Grp (Shp(X),) \Gamma_X \circ LConst_X(-) \;\; \simeq \;\; Grp_\infty \big( Shp(X) ,\, - \big)

Therefore, consider, for S 1,S 2Grpd S_1, S_2 \,\in\, Grpd_\infty, the following sequence of natural equivalences:

Grp (S 1,Γ XLConst X(S 2)) H /X(LConst X(S 1),LConst X(S 2)) H /X(X×Disc(S 1),X×Disc(S 2)) H(X×Disc(S 1),Disc(S 2)) Grp (Shp(X×Disc(S 1)),S 2) Grp (Shp(X)×S 1,S 2) Grp (S 1,Grp (Shp(X),S 2)) \begin{aligned} & \mathrm{Grp}_\infty \big( S_1 ,\, \Gamma_X \circ \mathrm{LConst}_X(S_2) \big) \\ & \;\simeq\; \mathbf{H}_{/X} \big( \mathrm{LConst}_X( S_1 ) ,\, \mathrm{LConst}_X(S_2) \big) \\ & \;\simeq\; \mathbf{H}_{/X} \big( X \times \mathrm{Disc}(S_1) ,\, X \times \mathrm{Disc}(S_2) \big) \\ & \;\simeq\; \mathbf{H} \big( X \times \mathrm{Disc}(S_1) ,\, \mathrm{Disc}(S_2) \big) \\ & \;\simeq\; \mathrm{Grp}_\infty \Big( \mathrm{Shp} \big( X \times \mathrm{Disc}(S_1) \big) ,\, S_2 \Big) \\ & \;\simeq\; \mathrm{Grp}_\infty \big( \mathrm{Shp}(X) \times S_1 ,\, S_2 \big) \\ & \;\simeq\; \mathrm{Grp}_\infty \Big( S_1 ,\, \mathrm{Grp}_\infty \big( \mathrm{Shp}(X) ,\, S_2 \big) \Big) \end{aligned}

Here the first four steps use the hom-equivalences of the above adjunctions. The second but last step uses cohesion, namely that the shape modality preserves finite homotopy products and is idempotent. The last step is the hom-equivalence for the cartesian monoidal-structure of ∞Grpd.

By the ( , 1 ) (\infty,1) -Yoneda lemma, this implies the equivalence (2) and hence the claim to be proven.

Shape of a topological space

To any topological space XX is associated the its \infty -category of \infty -sheaves (with respect its site of open subsets), which is an \infty -topos.

At least when XX is a compact Hausdorff space, then its strong shape in the classical sense of Mardešić & Segal 1971 does agree with the \infty -topos theoretic shape of its \infty -category of \infty -sheaves.

This fact must have motivated the terminology in Toën-Vezzosi 2002 and in Lurie 2009, Sec. 7.1.6; it is made explicit in Hoyois 2013, Rem. 2.13.

References

The notion of shape of an \infty-topos first appears, in essentially the form of the above Def. , in:

  • Bertrand Toën, Gabriele Vezzosi, Def. 5.3.2 in: Segal topoi and stacks over Segal categories, in: Proceedings of the Program Stacks, Intersection theory and Non-abelian Hodge Theory, MSRI, Berkeley, January-May 2002 (arXiv:math/0212330).

The concise formulation as ΓLConst()\Gamma \circ LConst(-), as in the above Def. , and discussion of relation to classical (strong) shape theory, of topological spaces, is due to

The further re-formulation as the image of the terminal object under the pro-left adjoint to LConstLConst is highlighted in

See also

Last revised on October 9, 2021 at 14:56:34. See the history of this page for a list of all contributions to it.