Clarify definition of 'enough points'
$begingroup$
I have checked the stacks project, Sketches of an Elephant, and Sheaves in Geometry and Logic but cannot find a satisfactory answer. If a site or topos has enough points, does that mean, that there is a SET of points, such that isomorphisms can be checked on that set of points? Or do we run into set-theoretic troubles.
topos-theory
asked Dec 17 '18 at 15:55
RoundthecornerRoundthecorner
1438
$endgroup$
add a comment |
$begingroup$
I have checked the stacks project, Sketches of an Elephant, and Sheaves in Geometry and Logic but cannot find a satisfactory answer. If a site or topos has enough points, does that mean, that there is a SET of points, such that isomorphisms can be checked on that set of points? Or do we run into set-theoretic troubles.
topos-theory
asked Dec 17 '18 at 15:55
RoundthecornerRoundthecorner
1438
$endgroup$
3
$begingroup$
I have no idea as to what this question is...
$endgroup$
– Don Thousand
Dec 17 '18 at 15:57
$begingroup$
Have you checked nLab? Does this definition satisfy your needs?
$endgroup$
– Christoph
Dec 17 '18 at 16:16
1
$begingroup$
@Rushabh: A topos is a kind of generalized space. Unlike ordinary topological spaces, a topos can be nontrivial even though it doesn't have any points. (Examples include the "topos of random binary sequences", the "topos of surjections $mathbb{N} to mathbb{R}$" and, in constructive mathematics, sometimes even the topos of prime filters of a given ring, which is just the topos-theoretic way of expressing the spectrum studied in algebraic-geometry.) For such toposes, to check whether some conditions hold, it isn't enough to check whether the conditions are ...
$endgroup$
– Ingo Blechschmidt
Dec 17 '18 at 17:31
1
$begingroup$
... satisfied at each point. However, in many important cases, a given topos actually does have enough points, so that this phenomenon doesn't occur. The question concerns a fine point about the definition of having enough points, on the set-theoretic subtletly that some collections, called proper classes, are in some sense too large to be sets.
$endgroup$
– Ingo Blechschmidt
Dec 17 '18 at 17:37
1
$begingroup$
There is much more about to toposes to say, including that they can be pictured as alternative mathematical universes suitable to do mathematics in, with intimate connections to the usual universe, but I'll stop here. You can learn more about toposes in this survey by Tom Leinster and more about proper classes here.
$endgroup$
– Ingo Blechschmidt
Dec 17 '18 at 17:37
add a comment |
$begingroup$
I have checked the stacks project, Sketches of an Elephant, and Sheaves in Geometry and Logic but cannot find a satisfactory answer. If a site or topos has enough points, does that mean, that there is a SET of points, such that isomorphisms can be checked on that set of points? Or do we run into set-theoretic troubles.
topos-theory
asked Dec 17 '18 at 15:55
RoundthecornerRoundthecorner
1438
$endgroup$
I have checked the stacks project, Sketches of an Elephant, and Sheaves in Geometry and Logic but cannot find a satisfactory answer. If a site or topos has enough points, does that mean, that there is a SET of points, such that isomorphisms can be checked on that set of points? Or do we run into set-theoretic troubles.
topos-theory
topos-theory
asked Dec 17 '18 at 15:55
RoundthecornerRoundthecorner
1438
asked Dec 17 '18 at 15:55
RoundthecornerRoundthecorner
1438
asked Dec 17 '18 at 15:55
RoundthecornerRoundthecorner
1438
asked Dec 17 '18 at 15:55
RoundthecornerRoundthecorner
1438
asked Dec 17 '18 at 15:55
RoundthecornerRoundthecorner
1438
1438
3
$begingroup$
I have no idea as to what this question is...
$endgroup$
– Don Thousand
Dec 17 '18 at 15:57
$begingroup$
Have you checked nLab? Does this definition satisfy your needs?
$endgroup$
– Christoph
Dec 17 '18 at 16:16
1
$begingroup$
@Rushabh: A topos is a kind of generalized space. Unlike ordinary topological spaces, a topos can be nontrivial even though it doesn't have any points. (Examples include the "topos of random binary sequences", the "topos of surjections $mathbb{N} to mathbb{R}$" and, in constructive mathematics, sometimes even the topos of prime filters of a given ring, which is just the topos-theoretic way of expressing the spectrum studied in algebraic-geometry.) For such toposes, to check whether some conditions hold, it isn't enough to check whether the conditions are ...
$endgroup$
– Ingo Blechschmidt
Dec 17 '18 at 17:31
1
$begingroup$
... satisfied at each point. However, in many important cases, a given topos actually does have enough points, so that this phenomenon doesn't occur. The question concerns a fine point about the definition of having enough points, on the set-theoretic subtletly that some collections, called proper classes, are in some sense too large to be sets.
$endgroup$
– Ingo Blechschmidt
Dec 17 '18 at 17:37
1
$begingroup$
There is much more about to toposes to say, including that they can be pictured as alternative mathematical universes suitable to do mathematics in, with intimate connections to the usual universe, but I'll stop here. You can learn more about toposes in this survey by Tom Leinster and more about proper classes here.
$endgroup$
– Ingo Blechschmidt
Dec 17 '18 at 17:37
add a comment |
3
$begingroup$
I have no idea as to what this question is...
$endgroup$
– Don Thousand
Dec 17 '18 at 15:57
$begingroup$
Have you checked nLab? Does this definition satisfy your needs?
$endgroup$
– Christoph
Dec 17 '18 at 16:16
1
$begingroup$
@Rushabh: A topos is a kind of generalized space. Unlike ordinary topological spaces, a topos can be nontrivial even though it doesn't have any points. (Examples include the "topos of random binary sequences", the "topos of surjections $mathbb{N} to mathbb{R}$" and, in constructive mathematics, sometimes even the topos of prime filters of a given ring, which is just the topos-theoretic way of expressing the spectrum studied in algebraic-geometry.) For such toposes, to check whether some conditions hold, it isn't enough to check whether the conditions are ...
$endgroup$
– Ingo Blechschmidt
Dec 17 '18 at 17:31
1
$begingroup$
... satisfied at each point. However, in many important cases, a given topos actually does have enough points, so that this phenomenon doesn't occur. The question concerns a fine point about the definition of having enough points, on the set-theoretic subtletly that some collections, called proper classes, are in some sense too large to be sets.
$endgroup$
– Ingo Blechschmidt
Dec 17 '18 at 17:37
1
$begingroup$
There is much more about to toposes to say, including that they can be pictured as alternative mathematical universes suitable to do mathematics in, with intimate connections to the usual universe, but I'll stop here. You can learn more about toposes in this survey by Tom Leinster and more about proper classes here.
$endgroup$
– Ingo Blechschmidt
Dec 17 '18 at 17:37
3
3
$begingroup$
I have no idea as to what this question is...
$endgroup$
– Don Thousand
Dec 17 '18 at 15:57
$begingroup$
I have no idea as to what this question is...
$endgroup$
– Don Thousand
Dec 17 '18 at 15:57
$begingroup$
Have you checked nLab? Does this definition satisfy your needs?
$endgroup$
– Christoph
Dec 17 '18 at 16:16
$begingroup$
Have you checked nLab? Does this definition satisfy your needs?
$endgroup$
– Christoph
Dec 17 '18 at 16:16
1
1
$begingroup$
@Rushabh: A topos is a kind of generalized space. Unlike ordinary topological spaces, a topos can be nontrivial even though it doesn't have any points. (Examples include the "topos of random binary sequences", the "topos of surjections $mathbb{N} to mathbb{R}$" and, in constructive mathematics, sometimes even the topos of prime filters of a given ring, which is just the topos-theoretic way of expressing the spectrum studied in algebraic-geometry.) For such toposes, to check whether some conditions hold, it isn't enough to check whether the conditions are ...
$endgroup$
– Ingo Blechschmidt
Dec 17 '18 at 17:31
$begingroup$
@Rushabh: A topos is a kind of generalized space. Unlike ordinary topological spaces, a topos can be nontrivial even though it doesn't have any points. (Examples include the "topos of random binary sequences", the "topos of surjections $mathbb{N} to mathbb{R}$" and, in constructive mathematics, sometimes even the topos of prime filters of a given ring, which is just the topos-theoretic way of expressing the spectrum studied in algebraic-geometry.) For such toposes, to check whether some conditions hold, it isn't enough to check whether the conditions are ...
$endgroup$
– Ingo Blechschmidt
Dec 17 '18 at 17:31
1
1
$begingroup$
... satisfied at each point. However, in many important cases, a given topos actually does have enough points, so that this phenomenon doesn't occur. The question concerns a fine point about the definition of having enough points, on the set-theoretic subtletly that some collections, called proper classes, are in some sense too large to be sets.
$endgroup$
– Ingo Blechschmidt
Dec 17 '18 at 17:37
$begingroup$
... satisfied at each point. However, in many important cases, a given topos actually does have enough points, so that this phenomenon doesn't occur. The question concerns a fine point about the definition of having enough points, on the set-theoretic subtletly that some collections, called proper classes, are in some sense too large to be sets.
$endgroup$
– Ingo Blechschmidt
Dec 17 '18 at 17:37
1
1
$begingroup$
There is much more about to toposes to say, including that they can be pictured as alternative mathematical universes suitable to do mathematics in, with intimate connections to the usual universe, but I'll stop here. You can learn more about toposes in this survey by Tom Leinster and more about proper classes here.
$endgroup$
– Ingo Blechschmidt
Dec 17 '18 at 17:37
$begingroup$
There is much more about to toposes to say, including that they can be pictured as alternative mathematical universes suitable to do mathematics in, with intimate connections to the usual universe, but I'll stop here. You can learn more about toposes in this survey by Tom Leinster and more about proper classes here.
$endgroup$
– Ingo Blechschmidt
Dec 17 '18 at 17:37
add a comment |
1 Answer
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$begingroup$
In the usual usage, for a topos to have enough points it is not required that there is a set of points such that isomorphisms can be detected by this set of points. It's enough if the full class of points does.
That said, it is a theorem that for Grothendieck toposes, the collection of all points detects isomorphisms if and only if there is a set of points which do so. This theorem is for instance stated on slide 3 of these slides by Olivia Caramello, and a proof is contained in Corollary C2.2.12 of the Elephant.
Finally, let me make a personal remark: As long as we don't precisely specify which foundations we work in, the question (and this answer) have a somewhat ill-defined flavor. Mike Shulman has a nice survey on available set-theoretic foundations, and then there are also foundations which are not set-theoretical in nature.
Addendum. Here is a concrete example. The ring classifier, that is the classifying topos of the theory of rings, that is the category $[mathrm{Ring}_{mathrm{fp}},mathrm{Set}]$ of $mathrm{Set}$-valued functors on the category of finitely presented rings, has a proper class of points, namely all rings. But for detecting isomorphisms it suffices to check at the finitely presented rings (which are incidentally precisely the "essential points" in the technical sense of essential geometric morphism), of which there is still a proper class, but we can restrict to some small, even countable skeleton of those.
answered Dec 17 '18 at 17:27
Ingo BlechschmidtIngo Blechschmidt
1,375815
$endgroup$
$begingroup$
Thank you. To be perfectly honest I am sometimes confused about the foundational stuff. The reference is much appreciated
$endgroup$
– Roundthecorner
Dec 18 '18 at 9:53
add a comment |
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$begingroup$
In the usual usage, for a topos to have enough points it is not required that there is a set of points such that isomorphisms can be detected by this set of points. It's enough if the full class of points does.
That said, it is a theorem that for Grothendieck toposes, the collection of all points detects isomorphisms if and only if there is a set of points which do so. This theorem is for instance stated on slide 3 of these slides by Olivia Caramello, and a proof is contained in Corollary C2.2.12 of the Elephant.
Finally, let me make a personal remark: As long as we don't precisely specify which foundations we work in, the question (and this answer) have a somewhat ill-defined flavor. Mike Shulman has a nice survey on available set-theoretic foundations, and then there are also foundations which are not set-theoretical in nature.
Addendum. Here is a concrete example. The ring classifier, that is the classifying topos of the theory of rings, that is the category $[mathrm{Ring}_{mathrm{fp}},mathrm{Set}]$ of $mathrm{Set}$-valued functors on the category of finitely presented rings, has a proper class of points, namely all rings. But for detecting isomorphisms it suffices to check at the finitely presented rings (which are incidentally precisely the "essential points" in the technical sense of essential geometric morphism), of which there is still a proper class, but we can restrict to some small, even countable skeleton of those.
answered Dec 17 '18 at 17:27
Ingo BlechschmidtIngo Blechschmidt
1,375815
$endgroup$
$begingroup$
Thank you. To be perfectly honest I am sometimes confused about the foundational stuff. The reference is much appreciated
$endgroup$
– Roundthecorner
Dec 18 '18 at 9:53
add a comment |
$begingroup$
In the usual usage, for a topos to have enough points it is not required that there is a set of points such that isomorphisms can be detected by this set of points. It's enough if the full class of points does.
That said, it is a theorem that for Grothendieck toposes, the collection of all points detects isomorphisms if and only if there is a set of points which do so. This theorem is for instance stated on slide 3 of these slides by Olivia Caramello, and a proof is contained in Corollary C2.2.12 of the Elephant.
Finally, let me make a personal remark: As long as we don't precisely specify which foundations we work in, the question (and this answer) have a somewhat ill-defined flavor. Mike Shulman has a nice survey on available set-theoretic foundations, and then there are also foundations which are not set-theoretical in nature.
Addendum. Here is a concrete example. The ring classifier, that is the classifying topos of the theory of rings, that is the category $[mathrm{Ring}_{mathrm{fp}},mathrm{Set}]$ of $mathrm{Set}$-valued functors on the category of finitely presented rings, has a proper class of points, namely all rings. But for detecting isomorphisms it suffices to check at the finitely presented rings (which are incidentally precisely the "essential points" in the technical sense of essential geometric morphism), of which there is still a proper class, but we can restrict to some small, even countable skeleton of those.
answered Dec 17 '18 at 17:27
Ingo BlechschmidtIngo Blechschmidt
1,375815
$endgroup$
$begingroup$
Thank you. To be perfectly honest I am sometimes confused about the foundational stuff. The reference is much appreciated
$endgroup$
– Roundthecorner
Dec 18 '18 at 9:53
add a comment |
$begingroup$
In the usual usage, for a topos to have enough points it is not required that there is a set of points such that isomorphisms can be detected by this set of points. It's enough if the full class of points does.
That said, it is a theorem that for Grothendieck toposes, the collection of all points detects isomorphisms if and only if there is a set of points which do so. This theorem is for instance stated on slide 3 of these slides by Olivia Caramello, and a proof is contained in Corollary C2.2.12 of the Elephant.
Finally, let me make a personal remark: As long as we don't precisely specify which foundations we work in, the question (and this answer) have a somewhat ill-defined flavor. Mike Shulman has a nice survey on available set-theoretic foundations, and then there are also foundations which are not set-theoretical in nature.
Addendum. Here is a concrete example. The ring classifier, that is the classifying topos of the theory of rings, that is the category $[mathrm{Ring}_{mathrm{fp}},mathrm{Set}]$ of $mathrm{Set}$-valued functors on the category of finitely presented rings, has a proper class of points, namely all rings. But for detecting isomorphisms it suffices to check at the finitely presented rings (which are incidentally precisely the "essential points" in the technical sense of essential geometric morphism), of which there is still a proper class, but we can restrict to some small, even countable skeleton of those.
answered Dec 17 '18 at 17:27
Ingo BlechschmidtIngo Blechschmidt
1,375815
$endgroup$
In the usual usage, for a topos to have enough points it is not required that there is a set of points such that isomorphisms can be detected by this set of points. It's enough if the full class of points does.
That said, it is a theorem that for Grothendieck toposes, the collection of all points detects isomorphisms if and only if there is a set of points which do so. This theorem is for instance stated on slide 3 of these slides by Olivia Caramello, and a proof is contained in Corollary C2.2.12 of the Elephant.
Finally, let me make a personal remark: As long as we don't precisely specify which foundations we work in, the question (and this answer) have a somewhat ill-defined flavor. Mike Shulman has a nice survey on available set-theoretic foundations, and then there are also foundations which are not set-theoretical in nature.
Addendum. Here is a concrete example. The ring classifier, that is the classifying topos of the theory of rings, that is the category $[mathrm{Ring}_{mathrm{fp}},mathrm{Set}]$ of $mathrm{Set}$-valued functors on the category of finitely presented rings, has a proper class of points, namely all rings. But for detecting isomorphisms it suffices to check at the finitely presented rings (which are incidentally precisely the "essential points" in the technical sense of essential geometric morphism), of which there is still a proper class, but we can restrict to some small, even countable skeleton of those.
answered Dec 17 '18 at 17:27
Ingo BlechschmidtIngo Blechschmidt
1,375815
edited Dec 18 '18 at 11:50
answered Dec 17 '18 at 17:27
Ingo BlechschmidtIngo Blechschmidt
1,375815
answered Dec 17 '18 at 17:27
Ingo BlechschmidtIngo Blechschmidt
1,375815
answered Dec 17 '18 at 17:27
Ingo BlechschmidtIngo Blechschmidt
1,375815
1,375815
$begingroup$
Thank you. To be perfectly honest I am sometimes confused about the foundational stuff. The reference is much appreciated
$endgroup$
– Roundthecorner
Dec 18 '18 at 9:53
add a comment |
$begingroup$
Thank you. To be perfectly honest I am sometimes confused about the foundational stuff. The reference is much appreciated
$endgroup$
– Roundthecorner
Dec 18 '18 at 9:53
$begingroup$
Thank you. To be perfectly honest I am sometimes confused about the foundational stuff. The reference is much appreciated
$endgroup$
– Roundthecorner
Dec 18 '18 at 9:53
$begingroup$
Thank you. To be perfectly honest I am sometimes confused about the foundational stuff. The reference is much appreciated
$endgroup$
– Roundthecorner
Dec 18 '18 at 9:53
add a comment |
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3
$begingroup$
I have no idea as to what this question is...
$endgroup$
– Don Thousand
Dec 17 '18 at 15:57
$begingroup$
Have you checked nLab? Does this definition satisfy your needs?
$endgroup$
– Christoph
Dec 17 '18 at 16:16
1
$begingroup$
@Rushabh: A topos is a kind of generalized space. Unlike ordinary topological spaces, a topos can be nontrivial even though it doesn't have any points. (Examples include the "topos of random binary sequences", the "topos of surjections $mathbb{N} to mathbb{R}$" and, in constructive mathematics, sometimes even the topos of prime filters of a given ring, which is just the topos-theoretic way of expressing the spectrum studied in algebraic-geometry.) For such toposes, to check whether some conditions hold, it isn't enough to check whether the conditions are ...
$endgroup$
– Ingo Blechschmidt
Dec 17 '18 at 17:31
1
$begingroup$
... satisfied at each point. However, in many important cases, a given topos actually does have enough points, so that this phenomenon doesn't occur. The question concerns a fine point about the definition of having enough points, on the set-theoretic subtletly that some collections, called proper classes, are in some sense too large to be sets.
$endgroup$
– Ingo Blechschmidt
Dec 17 '18 at 17:37
1
$begingroup$
There is much more about to toposes to say, including that they can be pictured as alternative mathematical universes suitable to do mathematics in, with intimate connections to the usual universe, but I'll stop here. You can learn more about toposes in this survey by Tom Leinster and more about proper classes here.
$endgroup$
– Ingo Blechschmidt
Dec 17 '18 at 17:37