nLab
shape of an (infinity,1)-topos

Idea

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

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

Definition

Definition

The composite (∞,1)-functor

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

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

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

Here

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

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

• $Y$ is the (∞,1)-Yoneda embedding;

• $\mathrm{Func}(-,-)$ is the (∞,1)-category of (∞,1)-functors;

• $\mathrm{AccLex}(-,-)\subset (\mathrm{\infty },1)\mathrm{Func}(-,-)$ 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.

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

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

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

Properties

Notice that for every (∞,1)-topos $H$ there is a unique geometric morphism

$$(\mathrm{LConst}⊣\Gamma ):H\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{\Gamma }{\to }}\mathrm{\infty }\mathrm{Grpd}$$(LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd

where ∞Grpd is the $(\mathrm{\infty },1)$-topos of ∞-groupoids, $\Gamma$ is the global sections (∞,1)-functor and $\mathrm{LConst}$ is the constant ∞-stack functor.

Proposition

The shape of $H$ is the composite functor

$$\Pi (H):=\Gamma \circ \mathrm{LConst}:\mathrm{\infty }\mathrm{Grpd}\stackrel{\mathrm{LConst}}{\to }H\stackrel{\Gamma }{\to }\mathrm{\infty }\mathrm{Grpd}$$\Pi(\mathbf{H}) := \Gamma \circ LConst \;\;:\;\; \infty Grpd \stackrel{LConst}{\to} \mathbf{H} \stackrel{\;\;\Gamma\;\;}{\to} \infty Grpd

regarded as an object

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

Proof

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

$$\mathrm{Func}(X,\mathrm{\infty }\mathrm{Grpd})\simeq \mathrm{\infty }\mathrm{Grpd}/X$$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 $\mathrm{Hom}(*,-):[X,\mathrm{\infty }\mathrm{Grpd}]\to \mathrm{\infty }\mathrm{Grpd}$ is the canonical projection $\mathrm{\infty }\mathrm{Grpd}/X\to \mathrm{\infty }\mathrm{Grpd}$. This means that it is an etale geometric morphism. So for any geometric morphism $f:H\to [X,\mathrm{\infty }\mathrm{Grpd}]$ we have a system of adjoint (∞,1)-functors

$$(\mathrm{LConst}⊣\Gamma ):H\stackrel{\stackrel{f^{*}}{\leftarrow }}{\underset{f_{*}}{\to }}\mathrm{\infty }\mathrm{Grpd}/X\stackrel{\stackrel{{\pi }^{*}}{\leftarrow }}{\underset{{\pi }_{*}}{\to }}\mathrm{\infty }\mathrm{Grpd}.$$(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 $\mathrm{\infty }\mathrm{Grpd}/X$ there is a canonical morphism

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

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

$$*\to f^{*}{\pi }^{*}X=\mathrm{LConst}X$$* \to f^* \pi^* X = LConst X

in $H$.

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

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

One checks that these constructions establish an equivalence

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

Using this, we see that

$$\begin{array}{rl}\Pi (H):X\mapsto & (\mathrm{\infty },1)\mathrm{Topos}(H,X)\\ & \simeq H(*,\mathrm{LConst}X)\\ & \simeq H(\mathrm{LConst}*,\mathrm{LConst}X)\\ & \simeq \mathrm{\infty }\mathrm{Grpd}(*,\Gamma \mathrm{LConst}X)\\ & \simeq \Gamma \mathrm{LConst}X\end{array}.$$\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 $\Pi (H):\mathrm{\infty }\mathrm{Grpd}\to \mathrm{\infty }\mathrm{Grpd}$ does preserve finite $(\mathrm{\infty },1)$-limits, since $\Gamma$ preserves all limits and $\mathrm{LConst}$ is a left exact functor. It also shows that it is accessible, since $\Gamma$ and $\mathrm{LConst}$ are both accessible.

Examples

Shape of a locally $\mathrm{\infty }$-connected topos

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

$$\begin{array}{rl}\mathrm{Shape}(H)(A)& =\Gamma (\mathrm{LConst}(A))\\ & ={\mathrm{Hom}}_{\mathrm{\infty }\mathrm{Grpd}}(*,\Gamma (\mathrm{LConst}(A)))\\ & ={\mathrm{Hom}}_H(\mathrm{LConst}(*),\mathrm{LConst}(A))\\ & ={\mathrm{Hom}}_H(*,\mathrm{LConst}(A))\\ & ={\mathrm{Hom}}_{\mathrm{\infty }\mathrm{Grpd}}(\Pi (*),A).\end{array}$$\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 $\Pi (*)$ as “the fundamental ∞-groupoid of $H$” — which is reasonable since when $H=\mathrm{Sh}(X)$ consists of sheaves on a locally contractible topological space $X$, ${\Pi }_H(*)$ is equivalent to the usual fundamental ∞-groupoid of $X$ — then we can regard the shape of an $(\mathrm{\infty },1)$-topos as a generalized version of the “homotopy $\mathrm{\infty }$-groupoid” which nevertheless makes sense even for non-locally-contractible toposes, by taking values in the larger category of “pro-$\mathrm{\infty }$-groupoids.”

It follows also that $H$ 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 $(\mathrm{\infty },1)$-topos theoretic shape of ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(X)$ relates to the ordinary shape-theoretic strong shape of the topological space $X$ see shape theory.

Shape of an essential retract

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

Observation

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

$$(\mathrm{Id}⊣\mathrm{Id})\simeq B\stackrel{\stackrel{f_{!}}{\leftarrow }}{\underset{f^{*}}{\to }}H\stackrel{\stackrel{f^{*}}{\leftarrow }}{\underset{f_{*}}{\to }}B.$$(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 $B$ is equivalent to that of $H$.

Proof

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

$$\begin{array}{rl}(\mathrm{\infty }\mathrm{Grpd}\stackrel{{\mathrm{LConst}}_B}{\to }B\stackrel{{\Gamma }_B}{\to }\mathrm{\infty }\mathrm{Grpd})& \simeq (\mathrm{\infty }\mathrm{Grpd}\stackrel{{\mathrm{LConst}}_B}{\to }B\stackrel{f^{*}}{\to }H\stackrel{f_{*}}{\to }B\stackrel{{\Gamma }_B}{\to }\mathrm{\infty }\mathrm{Grpd})\\ & \simeq (\mathrm{\infty }\mathrm{Grpd}\stackrel{{\mathrm{LConst}}_H}{\to }H\stackrel{{\Gamma }_H}{\to }\mathrm{\infty }\mathrm{Grpd})\end{array}.$$\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

• locally ∞-connected or ∞-connected (∞,1)-topos

• and hence also every cohesive (∞,1)-topos

over $\mathrm{\infty }\mathrm{Grpd}$ has the shape of the point.

Proof

By definition $H$ is $\mathrm{\infty }$-connected if the constant ∞-stack inverse image $f^{*}=L\mathrm{Const}$ 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 $f_{!}{f}^{*}\to \mathrm{Id}$ is an equivalence.

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

Related concepts

• shape of an $(\mathrm{\infty },1)$-topos

• coshape of an (∞,1)-topos

References

The definition of shape of $(\mathrm{\infty },1)$-toposes as $\Gamma \circ \mathrm{LConst}$ is due to

• Bertrand Toen, Gabriele Vezzosi, 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).

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

• Jacob Lurie, Higher Topos Theory