### Notions of Space

#### Posted by Urs Schreiber

Today is my turn in our Seminar on A Survey of Elliptic Cohomology.

I attempted to write a survey of some central ideas in Jacob Lurie’s Structured Spaces.

You can find it here: Notions of Space.

You may think of this post also as a continuation of our discussion about Comparative Smootheology I II III.

## Re: Notions of Space

That’s very nice! I have two questions:

Presumably the $(\infty,1)$-topos of (small) $(\infty,1)$-sheaves on $\mathcal{G}$ itself is the classifying (∞,1)-topos for $\mathcal{G}$-structures?

In the desired idea for a “concrete $\infty$-stack” $X(U) \subset \mathcal{L}Top(\mathcal{G})^{op}(Sh_{\infty}(U),\mathcal{X})$ I am a little confused, because

a priori$X(U)$ is an $\infty$-groupoid (since $X$ is a presheaf on $\mathcal{G}$ with values in $\infty$-groupoids), while in general the morphisms between two $(\infty,1)$-toposes form an $(\infty,1)$-category, correct? Do we require that $\mathcal{X}$ is a “groupoidal” $(\infty,1)$-topos, or at least that $X$ lives in its “groupoidal” part?