Return to Answer

add reference link

Source Link

edited Jun 24, 2018 at 15:48

13.6k
3
52
90

While I'm sure a hocolim solution also exists (and works in general), one can avoid it, at least for compact topological spaces, by using open covers which are indexed by the points of your space $X$ (this idea goes back at least to Godement). These are called "rigid open covers" by Carlsson--PedersenCarlsson–Pedersen, who make the following definition:

Definition: A rigid open cover of a compact Hausdorff space $X$ consists of open sets $\{U_x\subseteq X\}_{x\in X}$ such that $x\in U_x$, $\overline{\{x:U_x=U\}}\subseteq U$, and $\#\{U:U=U_x\text{ for some }x\}<\infty$.

Now we only consider refinements between rigid open covers which are the identity map on the index set (and hence there is at most one refinement between any pair of rigid open covers). Thus the Cech complexes form an honest direct system, and one can simply take the direct limit of complexes.

I don't know the maximum generality in which one can make such a construction, though probably the above generalizes at least to the locally compact Hausdorff setting.

While I'm sure a hocolim solution also exists (and works in general), one can avoid it, at least for compact topological spaces, by using open covers which are indexed by the points of your space $X$ (this idea goes back at least to Godement). These are called "rigid open covers" by Carlsson--Pedersen, who make the following definition:

Definition: A rigid open cover of a compact Hausdorff space $X$ consists of open sets $\{U_x\subseteq X\}_{x\in X}$ such that $x\in U_x$, $\overline{\{x:U_x=U\}}\subseteq U$, and $\#\{U:U=U_x\text{ for some }x\}<\infty$.

Now we only consider refinements between rigid open covers which are the identity map on the index set (and hence there is at most one refinement between any pair of rigid open covers). Thus the Cech complexes form an honest direct system, and one can simply take the direct limit of complexes.

I don't know the maximum generality in which one can make such a construction, though probably the above generalizes at least to the locally compact Hausdorff setting.

While I'm sure a hocolim solution also exists (and works in general), one can avoid it, at least for compact topological spaces, by using open covers which are indexed by the points of your space $X$ (this idea goes back at least to Godement). These are called "rigid open covers" by Carlsson–Pedersen, who make the following definition:

Definition: A rigid open cover of a compact Hausdorff space $X$ consists of open sets $\{U_x\subseteq X\}_{x\in X}$ such that $x\in U_x$, $\overline{\{x:U_x=U\}}\subseteq U$, and $\#\{U:U=U_x\text{ for some }x\}<\infty$.

Now we only consider refinements between rigid open covers which are the identity map on the index set (and hence there is at most one refinement between any pair of rigid open covers). Thus the Cech complexes form an honest direct system, and one can simply take the direct limit of complexes.

I don't know the maximum generality in which one can make such a construction, though probably the above generalizes at least to the locally compact Hausdorff setting.

Source Link

created Jun 24, 2018 at 13:25

18.7k
3
55
131

While I'm sure a hocolim solution also exists (and works in general), one can avoid it, at least for compact topological spaces, by using open covers which are indexed by the points of your space $X$ (this idea goes back at least to Godement). These are called "rigid open covers" by Carlsson--Pedersen, who make the following definition:

Definition: A rigid open cover of a compact Hausdorff space $X$ consists of open sets $\{U_x\subseteq X\}_{x\in X}$ such that $x\in U_x$, $\overline{\{x:U_x=U\}}\subseteq U$, and $\#\{U:U=U_x\text{ for some }x\}<\infty$.

Now we only consider refinements between rigid open covers which are the identity map on the index set (and hence there is at most one refinement between any pair of rigid open covers). Thus the Cech complexes form an honest direct system, and one can simply take the direct limit of complexes.

I don't know the maximum generality in which one can make such a construction, though probably the above generalizes at least to the locally compact Hausdorff setting.