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3 edited title

# DoesthisIsafunctorwhichisa sheaf propertyimplyanotheroneforopencoversandfiniteclosedcoversautomaticallyasheafforcoversbysimplices?

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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# Does this sheaf property imply another one?

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.