## November 4, 2009

### 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.

Posted at November 4, 2009 2:43 PM UTC

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2103

### Re: Notions of Space

That’s very nice! I have two questions:

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

2. 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?

Posted by: Mike Shulman on November 9, 2009 6:06 AM | Permalink | PGP Sig | Reply to this

### Re: Notions of Space

Thanks, Mike. Here a quick reply:

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

Exactly. That’s StSp prop 1.4.2.

in general the morphisms between two $(\infty,1)$-toposes form an $(\infty,1)$-category, correct?

In principle, that’s of course correct. But here we stay entirely in the world of $(\infty,1)$-categories and consider $\mathcal{L}Top(\mathcal{G})$ as an $(\infty,1)$-category itself, discarding the non-invertible 2-morphisms.

Posted by: Urs Schreiber on November 9, 2009 7:40 AM | Permalink | Reply to this

### Re: Notions of Space

Thanks Urs.

But here we stay entirely in the world of (∞,1)-categories and consider $\mathcal{L}Top(\mathcal{G})$ as an (∞,1)-category itself, discarding the non-invertible 2-morphisms.

There’s something funny about that, to my mind. Makes me think that we really want to be talking about $(\infty,\infty)$-topoi. (-:

Posted by: Mike Shulman on November 9, 2009 2:48 PM | Permalink | PGP Sig | Reply to this

### Re: Notions of Space

There’s something funny about that, to my mind. Makes me think that we really want to be talking about $(\infty,\infty)$-topoi. (-:

Sure. Eventually. I hope you are still working on that! :-)

Posted by: Urs Schreiber on November 9, 2009 2:54 PM | Permalink | Reply to this

Makes me think that we really want to be talking about $(\infty,\infty)$-topoi.