nLab shape of an (infinity,1)-topos

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 $X$ in the classical sense of shape theory.

In the special case that the $\infty$-topos is a slice of a cohesive $(\infty,1)$-topos $\mathbf{H}$ 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

$$Shp \;\colon\; (\infty,1)Topos \stackrel{Y}{\to} Func((\infty,1)Topos, \infty Grpd)^{op} \stackrel{Lex(PSh(-), \infty Grpd)}{\to} AccLex(\infty Grpd, \infty Grpd)^{op} \simeq Pro \infty Grpd$$

is the shape functor . Its value

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

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

Here:

• (∞,1)Topos is the (∞,1)-category of (∞,1)-toposes;

• ∞Grpd is the $(\infty,1)$-category of ∞-groupoids;

• $Y$ is the (∞,1)-Yoneda embedding;

• $Func(-,-)$ is the (∞,1)-category of (∞,1)-functors;

• $AccLex(-,-) \subset (\infty,1)Func(-,-)$ is the full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-functors on those which are left exact functors (preserve finite (∞,1)-limits) and also accessible.

• $PSh(-) : \infty Grpd \to (\infty,1)Topos$ is the functor that produces the (∞,1)-category of (∞,1)-presheaves $Func(X^{op}, \infty Grpd)$ on $X$ (equivalently on the equivalent opposite ∞-groupoid $X^{op}$);

• $Pro \infty Grpd$ is the (∞,1)-category of pro-objects in $\infty Grpd$.

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 $\mathbf{H}$ there is a unique geometric morphism

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

where

• ∞Grpd is the $(\infty,1)$-topos of ∞-groupoids

• $\Gamma$ is the global sections (∞,1)-functor

• $LConst$ is the constant ∞-stack functor.

Definition

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

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

regarded as a pro-$\infty$-groupoid:

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

Proposition

Def. and Def. are equivalent.

Proof

For $X \in$ ∞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

$$Func(X,\infty Grpd) \;\simeq\; \infty Grpd_{/X}$$

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

$$(LConst \dashv \Gamma) \;\colon\; \mathbf{H} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \infty Grpd/X \stackrel{\overset{\pi^*}{\leftarrow}}{\underset{\pi_*}{\to}} \infty Grpd \,.$$

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

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

$$(* \to \pi^* X) \;\coloneqq\; (X \stackrel{(Id,Id)}{\to} X \times X) \,.$$

The image of this under $f^*$ is (using that this preserves the terminal object) a morphism in $\mathbf{H}$ of the form

$$* \to f^* \pi^* X = LConst X \,.$$

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

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

One checks that these constructions establish an equivalence

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

Using this, we find the following sequence of equiavlences:

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

The composite of these is the equivalence to be shown.

Remark

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

Definition

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

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

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

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

Proposition

Def. is equivalent to Def.

Proof

Consider the following sequence of natural equivalences:

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

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 $\Gamma \dashv LConst$ . In the second but last step we use that $LConst$, 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 $\mathbf{H}$ is locally ∞-connected, in that $\LConst$ has an actual left $(\infty,1)$-functor $Shp$ (see at shape modality, in constrast to just a pro-left adjoint (1)) then the shape of $\mathbf{H}$ is $Shp(*) \in \infty Grpd \xhookrightarrow Prop \infty Grpd$ (under the embedding here):

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

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 $\infty Grpd \hookrightarrow Pro(\infty Grpd)$ (here)

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

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

Thus, if we regard $Shp(*)$ as “the fundamental ∞-groupoid of $\mathbf{H}$” — which is reasonable since when $\mathbf{H}=Sh(X)$ consists of sheaves on a locally contractible topological space $X$, $Shp_{\mathbf{H}}(*)$ is equivalent to the usual fundamental ∞-groupoid of $X$ — then we can regard the shape of an $(\infty,1)$-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 $\mathbf{H}$ is not only locally ∞-connected but also ∞-connected, then it has the shape of a point.

More generally:

Remark

Let $(f_! \dashv f^* \dashv f_*) : \mathbf{H} \xrightarrow{\;\;} \mathbf{B}$ be an essential geometric morphism of $(\infty,1)$-toposes that exhibits $\mathbf{B}$ as an essential retract of $\mathbf{H}$ in that

$$(Id \dashv Id) \;\; \simeq \;\; \mathbf{B} \underoverset {\underset{f^*}{\longrightarrow}} {\overset{f_!}{\longleftarrow}} {\;\;\;\bot\;\;\;} \mathbf{H} \underoverset {\underset{f_*}{\longrightarrow}} {\overset{f^*}{\longleftarrow}} {\;\;\;\bot\;\;\;} \mathbf{B} \,.$$

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

Proof

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

$$\begin{aligned} & (\infty Grpd \xrightarrow{\;LConst_{\mathbf{B}}\;} \mathbf{B} \xrightarrow{\;\Gamma_\mathbf{B}\;} \infty Grpd) \\ & \simeq (\infty Grpd \xrightarrow{\;LConst_{\mathbf{B}}\;} \mathbf{B} \xrightarrow{\;f^*\;} \mathbf{H} \xrightarrow{f_*} \mathbf{B} \xrightarrow{\Gamma_\mathbf{B}} \infty Grpd) \\ & \simeq (\infty Grpd \xrightarrow{LConst_\mathbf{H}} \mathbf{H} \xrightarrow{\Gamma_\mathbf{H}} \infty Grpd) \end{aligned} \,.$$

Example

(cohesive (∞,1)-topos has trivial shape)
Every

• $\infty$-topos which is both locally ∞-connected and $\infty$-connected,

• in particular every cohesive (∞,1)-topos

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

Proof

By definition $\mathbf{H}$ is $\infty$-connected if the constant ∞-stack inverse image $f^* = L Const$ 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 $f^*$ is a full and faithful (∞,1)-functor precisely if the counit $f_! f^* \to Id$ is an equivalence.

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

Remark

(trivial shape of gros $\infty$-toposes)
That cohesive $(\infty,1)$-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 $\mathbf{H}$ is a cohesive $(\infty,1)$-topos with shape modality

$$ʃ \;\; \colon \;\; \mathbf{H} \xrightarrow{\;Shp\;} \Grp_\infty \xrightarrow{\;Disc\;} \mathbf{H}$$

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

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

Proof

For $\mathbf{H}$ any $(\infty,1)$-topos and $X \,\in\, \mathbf{H}$ an object, the slice $(\infty,1)$-topos $\mathbf{H}_{/X}$ is related to $\mathbf{H}$ 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 $\mathbf{H}$:

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

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

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

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

Therefore, consider, for $S_1, S_2 \,\in\, Grpd_\infty$, the following sequence of natural equivalences:

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

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

Shape of a topological space

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

At least when $X$ 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.

Related concepts

• étale homotopy

• shape modality

• coshape of an (∞,1)-topos

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

• Jacob Lurie, Section 7.1.6 of: Higher Topos Theory, 2009

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

• Marc Hoyois, Def. 2.3 in: Higher Galois theory, Algebraic & Geometric Topology 17 1 (2017) 567-643 (arxiv/1506.07155, doi:10.2140/agt.2017.17.567)

See also

• Ilan Barnea, Yonatan Harpaz, Geoffroy Horel, Section 6 of: Pro-categories in homotopy theory, Algebraic & Geometric Topology 17 (2017) 567–643 (arxiv:1507.01564, doi:10.2140/agt.2017.17.567)