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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeDec 12th 2010
   • (edited Dec 12th 2010)
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   Author: Harry Gindi
   Format: MarkdownItexOver at [model structure for cartesian fibrations](http://ncatlab.org/nlab/show/model+structure+for+Cartesian+fibrations#as_a_quasitopos_35), one equips $\Delta^+$ with a Grothendieck coverage. Apparently, the separated presheaves with respect to this coverage are the marked simplicial sets. What are the sheaves? If someone could also explain what precisely is meant by equipment with a coverage, that would be helpful. I thought that the underlying category needs to have finite pullbacks to equip it with an honest coverage. In particular, this seems like a good time to use the subfunctor presentation of sieves to describe the Grothendieck topology, since the underlying category lacks pullbacks.

   Over at model structure for cartesian fibrations, one equips $\Delta^+$ with a Grothendieck coverage. Apparently, the separated presheaves with respect to this coverage are the marked simplicial sets. What are the sheaves?

   If someone could also explain what precisely is meant by equipment with a coverage, that would be helpful. I thought that the underlying category needs to have finite pullbacks to equip it with an honest coverage. In particular, this seems like a good time to use the subfunctor presentation of sieves to describe the Grothendieck topology, since the underlying category lacks pullbacks.

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeDec 12th 2010
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   Author: DavidRoberts
   Format: MarkdownItex>I thought that the underlying category needs to have finite pullbacks not necessarily - you only need pullbacks of covers to exist to have a pretopology (consider manifolds with the open covers as covers). For a coverage you don't even need that (only weak pullbacks) but at [[coverage]] a Grothendieck coverage is defined via sieves, so I'm not sure what you mean. I read the linked section at [model structure for cartesian fibrations](http://ncatlab.org/nlab/show/model+structure+for+Cartesian+fibrations#as_a_quasitopos_35) to mean that the coverage has only isomorphisms (=identities) as covers together with $c:[1] \to [1^+]$ (I've arbitrarily chosen the name $c$ for now). Now if you wanted to pull $c$ back along any map $k:[n] \to [1^+]$, we know that $k$ factors through $c$, so we need to first consider $k=c:[1] \to [1^+]$. But since $\Delta^+([1],[1^+]) = \{c\}$, there is a canonical pullback of $c$ along itself, namely $id_{[1]}$, which is a cover. And then all the rest is trivial, so it really is a pretopology.

   I thought that the underlying category needs to have finite pullbacks

   not necessarily - you only need pullbacks of covers to exist to have a pretopology (consider manifolds with the open covers as covers). For a coverage you don’t even need that (only weak pullbacks) but at coverage a Grothendieck coverage is defined via sieves, so I’m not sure what you mean. I read the linked section at model structure for cartesian fibrations to mean that the coverage has only isomorphisms (=identities) as covers together with $c:[1] \to [1^+]$ (I’ve arbitrarily chosen the name $c$ for now). Now if you wanted to pull $c$ back along any map $k:[n] \to [1^+]$, we know that $k$ factors through $c$, so we need to first consider $k=c:[1] \to [1^+]$. But since $\Delta^+([1],[1^+]) = \{c\}$, there is a canonical pullback of $c$ along itself, namely $id_{[1]}$, which is a cover. And then all the rest is trivial, so it really is a pretopology.

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeDec 13th 2010
   • (edited Dec 13th 2010)
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   Author: DavidRoberts
   Format: MarkdownItexI guess in light of the above, the only sheaf condition you'd need to check is the pasting along $c$ (defined in #2), the rest of the conditions are trivial, giving only presheaves. EDIT: and checking at the link you gave, it is obvious that sheaves are those marked simplicial sets with the maximum marking - all 1-simplices are in the 'marked subset'.

   I guess in light of the above, the only sheaf condition you’d need to check is the pasting along $c$ (defined in #2), the rest of the conditions are trivial, giving only presheaves.

   EDIT: and checking at the link you gave, it is obvious that sheaves are those marked simplicial sets with the maximum marking - all 1-simplices are in the ’marked subset’.

4. • CommentRowNumber4.
   • CommentAuthorHarry Gindi
   • CommentTimeDec 13th 2010
   • (edited Dec 13th 2010)
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   Author: Harry Gindi
   Format: MarkdownItexWuh oh, that's what I thought, and that's very not good.

   Wuh oh, that’s what I thought, and that’s very not good.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeDec 13th 2010
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   Author: Urs
   Format: MarkdownItexYes, what David writes is what is meant.

   Yes, what David writes is what is meant.

6. • CommentRowNumber6.
   • CommentAuthorHarry Gindi
   • CommentTimeDec 13th 2010
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   Author: Harry Gindi
   Format: MarkdownItexActually, nothing to be alarmed about. Everthing works out fine, since we're working over the base simplicial set $\Delta^0$. I suspect that things will work out alright in the overcategories as well.

   Actually, nothing to be alarmed about. Everthing works out fine, since we’re working over the base simplicial set $\Delta^0$. I suspect that things will work out alright in the overcategories as well.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeDec 13th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexCould you say what the aspect is that you mean when you first say "that's not very good" and now "nothing to be alarmed about"?

   Could you say what the aspect is that you mean when you first say “that’s not very good” and now “nothing to be alarmed about”?

8. • CommentRowNumber8.
   • CommentAuthorHarry Gindi
   • CommentTimeDec 13th 2010
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexWell, define a functor $\Delta^+\to Set^+_\Delta$ sending each object to the corresponding marked simplicial set (so this sends $[n]\mapsto (\Delta^n)^\flat$ and $[\tilde{1}]\mapsto (\Delta^1)^\sharp$. This lifts to a colimit preserving functor "realization" functor $Psh(\Delta^+)\to Set^+_\Delta$, which as as its adjoint functor the obvious "$Sing$" functor using the general formalism of the realization and the nerve functor). Now, it appears to me that $Sing(-)$ is the canonical inclusion of separated presheaves into presheaves. If that is the case, then that means that the "realization" functor is isomorphic to the plus construction. But that seems like it's really bad, because it destroys all of the marked anodyne structure that I want to put on $Psh(\Delta)$ (since double plusification sends our objects to , since all of the objects that we care about are separated anyway, and every single one of them gets sent to a maximally marked simplicial set.

   Well, define a functor $\Delta^+\to Set^+_\Delta$ sending each object to the corresponding marked simplicial set (so this sends $[n]\mapsto (\Delta^n)^\flat$ and $[\tilde{1}]\mapsto (\Delta^1)^\sharp$. This lifts to a colimit preserving functor “realization” functor $Psh(\Delta^+)\to Set^+_\Delta$, which as as its adjoint functor the obvious “$Sing$” functor using the general formalism of the realization and the nerve functor). Now, it appears to me that $Sing(-)$ is the canonical inclusion of separated presheaves into presheaves. If that is the case, then that means that the “realization” functor is isomorphic to the plus construction. But that seems like it’s really bad, because it destroys all of the marked anodyne structure that I want to put on $Psh(\Delta)$ (since double plusification sends our objects to , since all of the objects that we care about are separated anyway, and every single one of them gets sent to a maximally marked simplicial set.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeDec 13th 2010
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   Author: Urs
   Format: MarkdownItex> Well, define a functor $\Delta^+\to Set^+_\Delta$ sending each object to the corresponding marked simplicial set (so this sends $[n]\mapsto (\Delta^n)^\flat$ and $[\tilde{1}]\mapsto (\Delta^1)^\sharp$. Okay, that's the Yoneda embedding, which factors through separated presheaves for our subcanonical coverage. $\Delta^+ \hookrightarrow Set^+_{\Delta} \hookrightarrow PSh(\Delta^+)$. > This lifts to a colimit preserving functor "realization" functor $Psh(\Delta^+)\to Set^+_\Delta$, Yes, "concretization". > which as as its adjoint functor the obvious "$Sing$" functor using the general formalism of the realization and the nerve functor). Now, it appears to me that $Sing(-)$ is the canonical inclusion of separated presheaves into presheaves. Yes. > If that is the case, then that means that the "realization" functor is isomorphic to the plus construction. Yes. > But that seems like it's really bad, because it destroys all of the marked anodyne structure that I want to put on $Psh(\Delta)$ Hm , I am not following this sentence. But also I did not read everything you wrote above and in other threads in full detail.

   Well, define a functor $\Delta^+\to Set^+_\Delta$ sending each object to the corresponding marked simplicial set (so this sends $[n]\mapsto (\Delta^n)^\flat$ and $[\tilde{1}]\mapsto (\Delta^1)^\sharp$.

   Okay, that’s the Yoneda embedding, which factors through separated presheaves for our subcanonical coverage. $\Delta^+ \hookrightarrow Set^+_{\Delta} \hookrightarrow PSh(\Delta^+)$.

   This lifts to a colimit preserving functor “realization” functor $Psh(\Delta^+)\to Set^+_\Delta$,

   Yes, “concretization”.

   which as as its adjoint functor the obvious “$Sing$” functor using the general formalism of the realization and the nerve functor). Now, it appears to me that $Sing(-)$ is the canonical inclusion of separated presheaves into presheaves.

   Yes.

   If that is the case, then that means that the “realization” functor is isomorphic to the plus construction.

   Yes.

   But that seems like it’s really bad, because it destroys all of the marked anodyne structure that I want to put on $Psh(\Delta)$

   Hm , I am not following this sentence. But also I did not read everything you wrote above and in other threads in full detail.

10. • CommentRowNumber10.
    • CommentAuthorHarry Gindi
    • CommentTimeDec 13th 2010
    • (edited Dec 13th 2010)
    • PermaLink
    Author: Harry Gindi
    Format: MarkdownItexWell, doesn't this show that we can't reall have any sot of Quillen adjunction since the adjunction mangles the affine morphisms up really badly? For instance, given any minimally marked simplex $\Delta^n\in Psh(\Delta^+)$, this thing maps to a maximally marked simplex under the adjunction. The strange thing here is, I thought that the yoneda extension of the handpicked realization functor $\Delta^+\to Set_\Delta^+$ should extend $\Delta^n=y([n])$ and become a minimally marked simplicial set (where $[n]$ is taken to be an object of $\Delta^+$). However, since $\Delta^n$ is already a separated presheaf, that means that it must map to the maximally and minimally marked simplicies under the adjunction, but that doesn't make sense.

    Well, doesn’t this show that we can’t reall have any sot of Quillen adjunction since the adjunction mangles the affine morphisms up really badly?

    For instance, given any minimally marked simplex $\Delta^n\in Psh(\Delta^+)$, this thing maps to a maximally marked simplex under the adjunction.

    The strange thing here is, I thought that the yoneda extension of the handpicked realization functor $\Delta^+\to Set_\Delta^+$ should extend $\Delta^n=y([n])$ and become a minimally marked simplicial set (where $[n]$ is taken to be an object of $\Delta^+$). However, since $\Delta^n$ is already a separated presheaf, that means that it must map to the maximally and minimally marked simplicies under the adjunction, but that doesn’t make sense.

11. • CommentRowNumber11.
    • CommentAuthorHarry Gindi
    • CommentTimeDec 14th 2010
    • (edited Dec 14th 2010)
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    Author: Harry Gindi
    Format: MarkdownItexDear Urs, Here is the problem as I understand it: Consider the following functor: $T:\Delta^+\to Set_\Delta^+$ sending $[n]\mapsto (\Delta^n)^\flat$ for $n\geq 0$ and $[\tilde{1}]\mapsto (\Delta^1)^#$. We send the morphisms between them to the evident morphisms as well. Assuming that this construction is functorial, there exists a unique extension $|-|_T:Psh(\Delta^+)\to Set_\Delta^+$ so that the composition with the Yoneda embedding is equal (or at least we have an isomorphism of functors $|Y(-)|_T\cong T(-)$). Now, if this extension functor $|-|_T$ is the Grothendieck $+$-construction, notice the following: The representable presheaf $\Delta^n=Y([n])$ is separated! Here's the trouble: When we apply the $+$ construction to a presheaf that is already separated, this forms a sheaf. That means that $|\Delta^n|_T\cong (\Delta^n)^#$, but by the definition of $|-|_T$, this is evidently _not_ the case, since $T([n])$ was originally defined to be $(\Delta^n)^\flat$. What is the problem? My guess is that it is caused by some ambiguity in the definition of the $+$-construction, or whether or not the $+$-construction itself is the right "separification" functor to use. My guess is that it is not. I saw a construction in Vistoli's notes of a separafication functor defined as follows: Given any object $U$ in $C$, equip $F(U)$ with the following equivalence relation: $a~b$ if there exists a cover $\{c_i:U_i\to U\}_{i\in I}$ such that $c_i^*(a)=c_i^*(b)$ for every $i\in I$. Then we see that by the Grothendieck conditions, this equivalence relation lifts to an equivalence relation $R\rightrightarrows F$. Then we define $F^{sep}$ to be the colimit of the diagram (or the quotient by $R$). At least it looks to me like this separification is the identity on separated presheaves and is itself adjoint to the inclusion functor. That is, I think that the left adjoint of $\iota:SPsh\hookrightarrow Psh$ is given by $(-)^{sep}:Psh\to SPsh$. This does not contradict that $(-)^{++}$ is the left adjoint of the inclusion $\iota':Sh\hookrightarrow Psh$. At least, here's why I think so: Consider the following toy example: Let $C=[1]$ be the directed interval category, and equip it with the only nontrivial cover $0\to 1$. Then the separated presheaves for this category are the presheaves $F$ such that $s^*:F(1)\to F(0)$ is monic, right? Consider a morphism of presheaves $F\to G$ where $G$ is separated. Then this map factors uniquely through the quotient of $F$ by the equivalence relation on $F(1)$ such that $x~y$ if $s^*(x)=s^*(y)$ by the naturality of the diagram. That is to say, the "single plus" construction is not the reflector of the inclusion $\iota:SPsh\hookrightarrow Psh$.

    Dear Urs,

    Here is the problem as I understand it:

    Consider the following functor:

    $T:\Delta^+\to Set_\Delta^+$ sending $[n]\mapsto (\Delta^n)^\flat$ for $n\geq 0$ and $[\tilde{1}]\mapsto (\Delta^1)^#$. We send the morphisms between them to the evident morphisms as well.

    Assuming that this construction is functorial, there exists a unique extension $|-|_T:Psh(\Delta^+)\to Set_\Delta^+$ so that the composition with the Yoneda embedding is equal (or at least we have an isomorphism of functors $|Y(-)|_T\cong T(-)$).

    Now, if this extension functor $|-|_T$ is the Grothendieck $+$-construction, notice the following: The representable presheaf $\Delta^n=Y([n])$ is separated!

    Here’s the trouble: When we apply the $+$ construction to a presheaf that is already separated, this forms a sheaf. That means that $|\Delta^n|_T\cong (\Delta^n)^#$, but by the definition of $|-|_T$, this is evidently not the case, since $T([n])$ was originally defined to be $(\Delta^n)^\flat$.

    What is the problem?

    My guess is that it is caused by some ambiguity in the definition of the $+$-construction, or whether or not the $+$-construction itself is the right “separification” functor to use. My guess is that it is not. I saw a construction in Vistoli’s notes of a separafication functor defined as follows: Given any object $U$ in $C$, equip $F(U)$ with the following equivalence relation: $a~b$ if there exists a cover $\{c_i:U_i\to U\}_{i\in I}$ such that $c_i^*(a)=c_i^*(b)$ for every $i\in I$. Then we see that by the Grothendieck conditions, this equivalence relation lifts to an equivalence relation $R\rightrightarrows F$. Then we define $F^{sep}$ to be the colimit of the diagram (or the quotient by $R$).

    At least it looks to me like this separification is the identity on separated presheaves and is itself adjoint to the inclusion functor.

    That is, I think that the left adjoint of $\iota:SPsh\hookrightarrow Psh$ is given by $(-)^{sep}:Psh\to SPsh$. This does not contradict that $(-)^{++}$ is the left adjoint of the inclusion $\iota':Sh\hookrightarrow Psh$.

    At least, here’s why I think so:

    Consider the following toy example:

    Let $C=[1]$ be the directed interval category, and equip it with the only nontrivial cover $0\to 1$.

    Then the separated presheaves for this category are the presheaves $F$ such that $s^*:F(1)\to F(0)$ is monic, right?

    Consider a morphism of presheaves $F\to G$ where $G$ is separated. Then this map factors uniquely through the quotient of $F$ by the equivalence relation on $F(1)$ such that $x~y$ if $s^*(x)=s^*(y)$ by the naturality of the diagram.

    That is to say, the “single plus” construction is not the reflector of the inclusion $\iota:SPsh\hookrightarrow Psh$.

12. • CommentRowNumber12.
    • CommentAuthorHarry Gindi
    • CommentTimeDec 15th 2010
    • (edited Dec 15th 2010)
    • PermaLink
    Author: Harry Gindi
    Format: MarkdownItexAlright, Urs. Here's the problem: >> If that is the case, then that means that the "realization" functor is isomorphic to the plus construction. >Yes. This is not true, since the plus construction is _not_ left adjoint to the inclusion of separated presheaves. This left adjoint is given by the construction from Vistoli that I mentioned above. However, this makes me very happy, since if it were the case that the plus construction was the left adjoint, my entire plan would collapse into a state of disorder. However, right now, it looks like everything is going to work out alright, provided that a certain small lemma is true (separification preserves monomorphisms). If not, I may have to revise some hypotheses (by changing the cofibrations, although it looks like I'm probably safe for now).

    Alright, Urs. Here’s the problem:

    If that is the case, then that means that the “realization” functor is isomorphic to the plus construction.

    Yes.

    This is not true, since the plus construction is not left adjoint to the inclusion of separated presheaves. This left adjoint is given by the construction from Vistoli that I mentioned above.

    However, this makes me very happy, since if it were the case that the plus construction was the left adjoint, my entire plan would collapse into a state of disorder. However, right now, it looks like everything is going to work out alright, provided that a certain small lemma is true (separification preserves monomorphisms). If not, I may have to revise some hypotheses (by changing the cofibrations, although it looks like I’m probably safe for now).

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeDec 15th 2010
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    Author: Urs
    Format: MarkdownItexHi Harry, you are of course right, the "Yes" you quote was nonsense. Sorry for causing unnecessary confusion. To make up for my sins, I am working on bringing the entry [[separated presheaf]] into better shape.

    Hi Harry,

    you are of course right, the “Yes” you quote was nonsense. Sorry for causing unnecessary confusion. To make up for my sins, I am working on bringing the entry separated presheaf into better shape.

14. • CommentRowNumber14.
    • CommentAuthorHarry Gindi
    • CommentTimeDec 15th 2010
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    Author: Harry Gindi
    Format: MarkdownItexThanks! Much appreciated!

    Thanks! Much appreciated!