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edited Feb 22, 2022 at 5:27

If we have a compact Hausdorff space $S$, then my understanding is that the appropriate notion of the derived category of sheaves of condensed abelian groups is to consider the derived category $D_{\text{cond}}(S;\mathbb{Z})$ of sheaves of abelian groups on $*_{\text{proet}}/S$ (with the appropriate choice of the pro-etale site: compact Hausdorff spaces with finite jointly surjective covers). In this case, condensed cohomology of $S$ (which agrees with sheaf cohomology for discrete constant coefficients) is obtained using the usual formalism. Now, if $S$ is no-longer compact, but a compactly generated topological space, then there is a condensed set $\underline{S}$ associated to it. In this case, condensed cohomology of $S$ (which agrees with sheaf cohomology for discrete constant coefficients) is obtained using the usual formalism. Is there a site such that sheaves of abelian groups on it gives an appropriate notion of $D_{\text{cond}}(\underline{S},\mathbb{Z})$? For example, should one take covers to be given by compactly generated spaces that pullback under any compact Hausdorff subspace of $S$ to a covering by compact Hausdorff spaces finitely many of which are jointly surjective? One property I want it to have is that "condensed cohomology" $H^i_{\text{cond}}(\underline{S};A)$ for a condensed abelian group $A$ should be the one computed on this site. Basically, I want to know if there is a nice "condensed" site for topological spaces that are not necessarily compact.

If we have a compact Hausdorff space $S$, then my understanding is that the appropriate notion of the derived category of sheaves of condensed abelian groups is to consider the derived category $D_{\text{cond}}(S;\mathbb{Z})$ of sheaves of abelian groups on $*_{\text{proet}}/S$ (with the appropriate choice of the pro-etale site: compact Hausdorff spaces with finite jointly surjective covers). Now, if $S$ is no-longer compact, but a compactly generated topological space, then there is a condensed set $\underline{S}$ associated to it. In this case, condensed cohomology of $S$ (which agrees with sheaf cohomology for discrete constant coefficients) is obtained using the usual formalism. Is there a site such that sheaves of abelian groups on it gives an appropriate notion of $D_{\text{cond}}(\underline{S},\mathbb{Z})$? For example, should one take covers to be given by compactly generated spaces that pullback under any compact Hausdorff subspace of $S$ to a covering by compact Hausdorff spaces finitely many of which are jointly surjective? One property I want it to have is that "condensed cohomology" $H^i_{\text{cond}}(\underline{S};A)$ for a condensed abelian group $A$ should be the one computed on this site. Basically, I want to know if there is a nice "condensed" site for topological spaces that are not necessarily compact.

If we have a compact Hausdorff space $S$, then my understanding is that the appropriate notion of the derived category of sheaves of condensed abelian groups is to consider the derived category $D_{\text{cond}}(S;\mathbb{Z})$ of sheaves of abelian groups on $*_{\text{proet}}/S$ (with the appropriate choice of the pro-etale site: compact Hausdorff spaces with finite jointly surjective covers). In this case, condensed cohomology of $S$ (which agrees with sheaf cohomology for discrete constant coefficients) is obtained using the usual formalism. Now, if $S$ is no-longer compact, but a compactly generated topological space, then there is a condensed set $\underline{S}$ associated to it. Is there a site such that sheaves of abelian groups on it gives an appropriate notion of $D_{\text{cond}}(\underline{S},\mathbb{Z})$? For example, should one take covers to be given by compactly generated spaces that pullback under any compact Hausdorff subspace of $S$ to a covering by compact Hausdorff spaces finitely many of which are jointly surjective? One property I want it to have is that "condensed cohomology" $H^i_{\text{cond}}(\underline{S};A)$ for a condensed abelian group $A$ should be the one computed on this site. Basically, I want to know if there is a nice "condensed" site for topological spaces that are not necessarily compact.

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asked Feb 22, 2022 at 4:42

user447643

Appropriate notion of derived category over condensed set

If we have a compact Hausdorff space $S$, then my understanding is that the appropriate notion of the derived category of sheaves of condensed abelian groups is to consider the derived category $D_{\text{cond}}(S;\mathbb{Z})$ of sheaves of abelian groups on $*_{\text{proet}}/S$ (with the appropriate choice of the pro-etale site: compact Hausdorff spaces with finite jointly surjective covers). Now, if $S$ is no-longer compact, but a compactly generated topological space, then there is a condensed set $\underline{S}$ associated to it. In this case, condensed cohomology of $S$ (which agrees with sheaf cohomology for discrete constant coefficients) is obtained using the usual formalism. Is there a site such that sheaves of abelian groups on it gives an appropriate notion of $D_{\text{cond}}(\underline{S},\mathbb{Z})$? For example, should one take covers to be given by compactly generated spaces that pullback under any compact Hausdorff subspace of $S$ to a covering by compact Hausdorff spaces finitely many of which are jointly surjective? One property I want it to have is that "condensed cohomology" $H^i_{\text{cond}}(\underline{S};A)$ for a condensed abelian group $A$ should be the one computed on this site. Basically, I want to know if there is a nice "condensed" site for topological spaces that are not necessarily compact.