Hereditary topological axioms: from X to its etale space
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So, this a question that arose from reading the first few pages of Ramanan's (excellent) book "Global Analysis", which uses modern algebraic notions used mainly in Category theory, while avoiding any actual Category theory.
Let $X$ be some topological space, and let $mathcal F$ be some presheaf defined on $X$. Let us assume $X$ fulfills some topological axioms, say it is Hausdorff. Is there some necessary condition that $mathcal F$ must fulfill in order to make sure that the associated etale space (IE the disjoint union of all stalks $mathcal F _x$ for all $xin X$ together with the topology generated by the sets $tilde s (U)$, the germs of elements of the open sets of $X$) is also Hausdorff? Is it Hausdorff for any presheaf? What about other topological axioms?
sheaf-theory
asked Sep 5 at 8:58
Uri George Peterzil
578
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up vote
0
down vote
favorite
So, this a question that arose from reading the first few pages of Ramanan's (excellent) book "Global Analysis", which uses modern algebraic notions used mainly in Category theory, while avoiding any actual Category theory.
Let $X$ be some topological space, and let $mathcal F$ be some presheaf defined on $X$. Let us assume $X$ fulfills some topological axioms, say it is Hausdorff. Is there some necessary condition that $mathcal F$ must fulfill in order to make sure that the associated etale space (IE the disjoint union of all stalks $mathcal F _x$ for all $xin X$ together with the topology generated by the sets $tilde s (U)$, the germs of elements of the open sets of $X$) is also Hausdorff? Is it Hausdorff for any presheaf? What about other topological axioms?
sheaf-theory
asked Sep 5 at 8:58
Uri George Peterzil
578
1
Certainly it's going to be a rare property. The étale space of continuous real-valued functions tends to be non-Hausdorff.
– Sofie Verbeek
Sep 12 at 15:16
1
Unravelling the definitions to show what it means for an étale space to be Hausdorff gives you a rather weird (tautological) criterion. On first sight it does not seem to have a decent intuitive interpretation.
– Sofie Verbeek
Sep 12 at 15:19
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
So, this a question that arose from reading the first few pages of Ramanan's (excellent) book "Global Analysis", which uses modern algebraic notions used mainly in Category theory, while avoiding any actual Category theory.
Let $X$ be some topological space, and let $mathcal F$ be some presheaf defined on $X$. Let us assume $X$ fulfills some topological axioms, say it is Hausdorff. Is there some necessary condition that $mathcal F$ must fulfill in order to make sure that the associated etale space (IE the disjoint union of all stalks $mathcal F _x$ for all $xin X$ together with the topology generated by the sets $tilde s (U)$, the germs of elements of the open sets of $X$) is also Hausdorff? Is it Hausdorff for any presheaf? What about other topological axioms?
sheaf-theory
asked Sep 5 at 8:58
Uri George Peterzil
578
So, this a question that arose from reading the first few pages of Ramanan's (excellent) book "Global Analysis", which uses modern algebraic notions used mainly in Category theory, while avoiding any actual Category theory.
Let $X$ be some topological space, and let $mathcal F$ be some presheaf defined on $X$. Let us assume $X$ fulfills some topological axioms, say it is Hausdorff. Is there some necessary condition that $mathcal F$ must fulfill in order to make sure that the associated etale space (IE the disjoint union of all stalks $mathcal F _x$ for all $xin X$ together with the topology generated by the sets $tilde s (U)$, the germs of elements of the open sets of $X$) is also Hausdorff? Is it Hausdorff for any presheaf? What about other topological axioms?
sheaf-theory
sheaf-theory
asked Sep 5 at 8:58
Uri George Peterzil
578
asked Sep 5 at 8:58
Uri George Peterzil
578
edited Sep 5 at 9:28
asked Sep 5 at 8:58
Uri George Peterzil
578
asked Sep 5 at 8:58
Uri George Peterzil
578
asked Sep 5 at 8:58
Uri George Peterzil
578
578
1
Certainly it's going to be a rare property. The étale space of continuous real-valued functions tends to be non-Hausdorff.
– Sofie Verbeek
Sep 12 at 15:16
1
Unravelling the definitions to show what it means for an étale space to be Hausdorff gives you a rather weird (tautological) criterion. On first sight it does not seem to have a decent intuitive interpretation.
– Sofie Verbeek
Sep 12 at 15:19
add a comment |Â
1
Certainly it's going to be a rare property. The étale space of continuous real-valued functions tends to be non-Hausdorff.
– Sofie Verbeek
Sep 12 at 15:16
1
Unravelling the definitions to show what it means for an étale space to be Hausdorff gives you a rather weird (tautological) criterion. On first sight it does not seem to have a decent intuitive interpretation.
– Sofie Verbeek
Sep 12 at 15:19
1
1
Certainly it's going to be a rare property. The étale space of continuous real-valued functions tends to be non-Hausdorff.
– Sofie Verbeek
Sep 12 at 15:16
Certainly it's going to be a rare property. The étale space of continuous real-valued functions tends to be non-Hausdorff.
– Sofie Verbeek
Sep 12 at 15:16
1
1
Unravelling the definitions to show what it means for an étale space to be Hausdorff gives you a rather weird (tautological) criterion. On first sight it does not seem to have a decent intuitive interpretation.
– Sofie Verbeek
Sep 12 at 15:19
Unravelling the definitions to show what it means for an étale space to be Hausdorff gives you a rather weird (tautological) criterion. On first sight it does not seem to have a decent intuitive interpretation.
– Sofie Verbeek
Sep 12 at 15:19
add a comment |Â
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1
Certainly it's going to be a rare property. The étale space of continuous real-valued functions tends to be non-Hausdorff.
– Sofie Verbeek
Sep 12 at 15:16
1
Unravelling the definitions to show what it means for an étale space to be Hausdorff gives you a rather weird (tautological) criterion. On first sight it does not seem to have a decent intuitive interpretation.
– Sofie Verbeek
Sep 12 at 15:19