Suppose $F:Top^{op}\rightarrow Set$ is a functor which is a sheaf for open coverings as well as for finite closed coverings, i.e. such that, apart from the usual sheaf property we also have that for every finite covering of a space $X$ by closed sets $\{U_i\}$ and every compatible familiy $f_i \in F(U_i)$ there is a unique $f \in F(X)$ which restricts to the $f_i$.

Does then $F$ also have the sheaf property for coverings of geometric realizations of simplicial sets by their simplices?

I.e. let $K$ be a simplicial set and let $D$ be the diagram in $sSet$ obtained from the comma category $\Delta \downarrow K$ by removing $K$ and the morphisms into it ($K$ is the colimit of this diagram in $sSet$). Now let $|D|$ be the geometric realization of $D$, which is a diagram of topological simplices in $Top$. Then the question is: If we have a compatible family of elements of $F$, for this diagram, can we glue them together to an element of $F(colim|D|)$?

Motivation: We would like to compare $F$ to the functor $Hom_{Top}(-,|Sing(F)|)$ where Sing(F) is the simplicial set $F(\Delta^{\bullet})$ obtained by applying $F$ to the cosimplicial object given by the $Hom_{Top}(-,|\Delta^n|)$ in $Set^{Top^{op}}$. If $F$ is itself representable, i.e. is a topological space $X$, then the two functors are weakly equivalent in a sense suitably extended from $Top$ to the functor category $Set^{Top^{op}}$. The same could be hoped for $F$, and in order to analyse the natural transformations between them one has to look at what they do at single spaces. Using some adjunction considerations one sees that one can restrict to spaces of the form $|K|$. But maps $Hom_{Top}(-,|K|) \rightarrow F$ in $Set^{Top^{op}}$ correspond to elements of $F(|K|)$ and the above sheaf property would then allow to look just at topological simplices.

Put differently (if this helps anything), I am asking whether the functor $Top \rightarrow Set^{Top^{op}} \rightarrow Sh$ given by the Yoneda embedding followed by sheafification for the Grothendieck topology generated by open covers and finite closed covers preserves the colimits of diagrams of simplices which arise in the above fashion...

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    $\begingroup$ Please try to give a more descriptive title. You have something like 200 or 250 characters for the title --- more than a tweet. For example, this comment fits in a title on MathOverflow. $\endgroup$ – Theo Johnson-Freyd Apr 25 '10 at 2:21
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    $\begingroup$ I only lconsidered a few examples but it seems that looking at the intersection pattern of the simplices of the barycentric subdivision (and using the sheaf property for closed coverings) would suffice. I know that there are situations where there is a difference between the first and second barycentric subdivisions so one might have to go to the simplices in the second subdivision. This would handle simplicial sets with only a finite number of non-degenerate simplices. The general case could then possibly be handled by using the open covering property using stars of finite subcomplexes. $\endgroup$ – Torsten Ekedahl Apr 25 '10 at 18:32
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    $\begingroup$ One thing that confuses me in the general case are scenarios of the following kind: Put a triangle into R^3, placing its base onto the x-axis, then rotate it around the x-axis. This gives uncountably many simplices all coinciding at one face, which occur for example in the simplicial set Sing(R^3). I have no idea how to get the sheaf property for such arrangements... $\endgroup$ – Peter Arndt Apr 26 '10 at 22:00
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    $\begingroup$ @Peter: Have you made any progress with this? $\endgroup$ – David Carchedi May 9 '12 at 2:45
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    $\begingroup$ No, I haven't thought about it for two years. Do you have an idea? $\endgroup$ – Peter Arndt May 9 '12 at 17:51

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