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1. • CommentRowNumber1.
   • CommentAuthorJon Beardsley
   • CommentTimeAug 4th 2013
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   Author: Jon Beardsley
   Format: MarkdownItexHi again! So wondering if I could bug somebody with a question about simplicial schemes. If one has a simplicial scheme, presumably there is a notion of "geometric realization"? Suppose then that my simplicial scheme is affine, i.e. I've started with a cosimplicial ring (....$\infty$-quantity? right Urs?....) and applied $Spec(-)$ levelwise. If geometrically realize such a simplicial scheme, do I get some kind of "spec" of the totalization of the cosimplicial ring? What sorts of things are totalized objects of cosimplicial rings anyway? Coloops on a cogroup object or something? Anyway, would love to get some input, though it might be a completely silly pursuit. Thanks!! -Jon

   Hi again!

   So wondering if I could bug somebody with a question about simplicial schemes. If one has a simplicial scheme, presumably there is a notion of “geometric realization”? Suppose then that my simplicial scheme is affine, i.e. I’ve started with a cosimplicial ring (….$\infty$-quantity? right Urs?….) and applied $Spec(-)$ levelwise. If geometrically realize such a simplicial scheme, do I get some kind of “spec” of the totalization of the cosimplicial ring? What sorts of things are totalized objects of cosimplicial rings anyway? Coloops on a cogroup object or something? Anyway, would love to get some input, though it might be a completely silly pursuit.

   Thanks!!

   -Jon

2. • CommentRowNumber2.
   • CommentAuthorDylan Wilson
   • CommentTimeAug 4th 2013
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   Author: Dylan Wilson
   Format: MarkdownItexThe category of schemes doesn't have arbitrary colimits, so to my knowledge there *isn't* really a "geometric realization" without passing to one of the many generalizations of schemes that exist now-adays (one of these generalizations is just... simplicial schemes, in which "realization" would be vacuous). Moreover, the inclusion of affine schemes inside all schemes does not preserve colimits, so even if the particular geometric realization happened to exist, there's no immediate guarantee that you'd get the same answer as taking the realization first inside affine schemes. Though, actually, the inclusion of affine schemes might preserve certain kinds of colimits, (we know, for example, it preserves finite coproducts), so you could be okay... I haven't thought about this.

   The category of schemes doesn’t have arbitrary colimits, so to my knowledge there isn’t really a “geometric realization” without passing to one of the many generalizations of schemes that exist now-adays (one of these generalizations is just… simplicial schemes, in which “realization” would be vacuous).

   Moreover, the inclusion of affine schemes inside all schemes does not preserve colimits, so even if the particular geometric realization happened to exist, there’s no immediate guarantee that you’d get the same answer as taking the realization first inside affine schemes. Though, actually, the inclusion of affine schemes might preserve certain kinds of colimits, (we know, for example, it preserves finite coproducts), so you could be okay… I haven’t thought about this.

3. • CommentRowNumber3.
   • CommentAuthorTim_Porter
   • CommentTimeAug 4th 2013
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   Author: Tim_Porter
   Format: MarkdownItexThere are mentions of étale realisation in the literature in the context of motives etc. I do not know that stuff well enough to compare.

   There are mentions of étale realisation in the literature in the context of motives etc. I do not know that stuff well enough to compare.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeAug 4th 2013
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   Author: zskoda
   Format: MarkdownItexWhy not consider simplicial schemes as special kinds of simplicial presheaves on $Aff$, so that one gets a realization in the codomain as a special kind of functor from $Aff^{op}$ to spaces ? This may be useful occasionally.

   Why not consider simplicial schemes as special kinds of simplicial presheaves on $Aff$, so that one gets a realization in the codomain as a special kind of functor from $Aff^{op}$ to spaces ? This may be useful occasionally.

5. • CommentRowNumber5.
   • CommentAuthorJon Beardsley
   • CommentTimeAug 5th 2013
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   Author: Jon Beardsley
   Format: MarkdownItexYeah, I'm thinking in terms of simplicial presheaves, so I guess my question is really, if you identify such a scheme with the associated presheaf, do you get any kind of represented object back? Moreover, if you start with an affine guy, do you get an affine guy back. I don't suppose it's particularly important to know, once one gets comfortable with the functor of points perspective and so forth, but it's just something I thought about. At the very least, there should be some kind of duality between totalizations of cosimplicial copresheaves and "realizations" of simplicial sheaves.

   Yeah, I’m thinking in terms of simplicial presheaves, so I guess my question is really, if you identify such a scheme with the associated presheaf, do you get any kind of represented object back? Moreover, if you start with an affine guy, do you get an affine guy back. I don’t suppose it’s particularly important to know, once one gets comfortable with the functor of points perspective and so forth, but it’s just something I thought about. At the very least, there should be some kind of duality between totalizations of cosimplicial copresheaves and “realizations” of simplicial sheaves.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeAug 5th 2013
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   Author: Urs
   Format: MarkdownItexI haven't really been following the discussion. But just one comment that springs to mind on the last comment: the geometric realization of simplicial sheaves is, if it exists, the derived left adjoint to the "constant" simplicial sheaf functor. See at _[[geometric realization of cohesive infinity-groupoids]]_.

   I haven’t really been following the discussion. But just one comment that springs to mind on the last comment:

   the geometric realization of simplicial sheaves is, if it exists, the derived left adjoint to the “constant” simplicial sheaf functor. See at geometric realization of cohesive infinity-groupoids.

7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeAug 6th 2013
   • (edited Aug 6th 2013)
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   Author: zskoda
   Format: MarkdownItex5: I do not understand the question: affine guys are representables. All schemes are locally representable sheaves on a Zariski topology on $Aff$. The passage between sheaf picture and locally ringed spaces is in Knudson's _Algebraic spaces_. Now, I do not understand if you really need totalization, if one works with sheaves of simplicial spaces on arbitrary site, do you really need to totalize ? 6, Urs: this is in infinity-category of simplicial sheaves, and 5 I think might want a geometric realization which will be equivalence at 1-categorical level instead. So, the question is valid at underived level already, that is why I said levelwise.

   5: I do not understand the question: affine guys are representables. All schemes are locally representable sheaves on a Zariski topology on $Aff$. The passage between sheaf picture and locally ringed spaces is in Knudson’s Algebraic spaces. Now, I do not understand if you really need totalization, if one works with sheaves of simplicial spaces on arbitrary site, do you really need to totalize ?

   6, Urs: this is in infinity-category of simplicial sheaves, and 5 I think might want a geometric realization which will be equivalence at 1-categorical level instead. So, the question is valid at underived level already, that is why I said levelwise.