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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

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Re: Notions of Space

That’s very nice! I have two questions:

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

  2. In the desired idea for a “concrete ∞\infty-stack” X(U)⊂ℒTop(𝒢) op(Sh ∞(U),𝒳)X(U) \subset \mathcal{L}Top(\mathcal{G})^{op}(Sh_{\infty}(U),\mathcal{X}) I am a little confused, because a priori X(U)X(U) is an ∞\infty-groupoid (since XX is a presheaf on 𝒢\mathcal{G} with values in ∞\infty-groupoids), while in general the morphisms between two (∞,1)(\infty,1)-toposes form an (∞,1)(\infty,1)-category, correct? Do we require that 𝒳\mathcal{X} is a “groupoidal” (∞,1)(\infty,1)-topos, or at least that XX 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 (∞,1)(\infty,1)-topos of (small) (∞,1)(\infty,1)-sheaves on 𝒢\mathcal{G} itself is the classifying (∞,1)(\infty,1)-topos for 𝒢\mathcal{G}-structures?

Exactly. That’s StSp prop 1.4.2.

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

In principle, that’s of course correct. But here we stay entirely in the world of (∞,1)(\infty,1)-categories and consider ℒTop(𝒢)\mathcal{L}Top(\mathcal{G}) as an (∞,1)(\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 ℒTop(𝒢)\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

Toperads

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

By the way: do you (or anyone else reading this) know about any attempts to consider categories of operad-valued sheaves, dendroidal-set-valued sheaves and the like? “Toperads”?

Posted by: Urs Schreiber on November 9, 2009 3:02 PM | Permalink | Reply to this

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