Contents

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

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

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.

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

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.

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