How is it possible to recover properties of compactness, connectedness, hausdorff, of topological space by...
1
$begingroup$
I will rephrase the title :
How is it possible to effectively recover/express all classical properties stated above of a topological space by using the category of presheaves of open subsets ?
general-topology category-theory topos-theory
asked May 26 '18 at 17:05
metameta
42
$endgroup$
add a comment |
1
$begingroup$
I will rephrase the title :
How is it possible to effectively recover/express all classical properties stated above of a topological space by using the category of presheaves of open subsets ?
general-topology category-theory topos-theory
asked May 26 '18 at 17:05
metameta
42
$endgroup$
$begingroup$
At best you can hope to recover the lattice of open subsets, which I think you can do. This recovers the original space if the space is sober but not in general.
$endgroup$
– Qiaochu Yuan
May 26 '18 at 17:15
$begingroup$
A presheaf is a contravariant functor from $O(X)$ to what category? Or what else do you mean by the category of presheaves?
$endgroup$
– Henno Brandsma
May 27 '18 at 6:54
$begingroup$
the set of all open subsets of a topological space is a poset, hence form a category, a functor from this category to the category of sets is a presheaf but it could be any category as the target. I just wanted to know how to express these classical properties of a topological space but now expressed in categorial formalism
$endgroup$
– meta
May 27 '18 at 23:04
1
$begingroup$
by category of presheaves I mean a category where the objects are functors from the category formed by all open subsets to another category
$endgroup$
– meta
May 28 '18 at 3:54
$begingroup$
If you were referring to the category of sheaves instead of the category of presheaves, it would work as follows: The set of subobjects of the terminal object of the category of sheaves is canonically a poset, and in fact it is isomorphic to the poset of opens of your given space. Hence you can recover the poset of opens. This is enough to, for instance, detect compactness and locality. In general, you'll need the sobrity condition Qiaochu already mentioned.
$endgroup$
– Ingo Blechschmidt
Dec 14 '18 at 22:43
add a comment |
1
1
1
$begingroup$
I will rephrase the title :
How is it possible to effectively recover/express all classical properties stated above of a topological space by using the category of presheaves of open subsets ?
general-topology category-theory topos-theory
asked May 26 '18 at 17:05
metameta
42
$endgroup$
I will rephrase the title :
How is it possible to effectively recover/express all classical properties stated above of a topological space by using the category of presheaves of open subsets ?
general-topology category-theory topos-theory
general-topology category-theory topos-theory
asked May 26 '18 at 17:05
metameta
42
asked May 26 '18 at 17:05
metameta
42
edited Dec 15 '18 at 15:33
meta
asked May 26 '18 at 17:05
metameta
42
asked May 26 '18 at 17:05
metameta
42
asked May 26 '18 at 17:05
metameta
42
42
$begingroup$
At best you can hope to recover the lattice of open subsets, which I think you can do. This recovers the original space if the space is sober but not in general.
$endgroup$
– Qiaochu Yuan
May 26 '18 at 17:15
$begingroup$
A presheaf is a contravariant functor from $O(X)$ to what category? Or what else do you mean by the category of presheaves?
$endgroup$
– Henno Brandsma
May 27 '18 at 6:54
$begingroup$
the set of all open subsets of a topological space is a poset, hence form a category, a functor from this category to the category of sets is a presheaf but it could be any category as the target. I just wanted to know how to express these classical properties of a topological space but now expressed in categorial formalism
$endgroup$
– meta
May 27 '18 at 23:04
1
$begingroup$
by category of presheaves I mean a category where the objects are functors from the category formed by all open subsets to another category
$endgroup$
– meta
May 28 '18 at 3:54
$begingroup$
If you were referring to the category of sheaves instead of the category of presheaves, it would work as follows: The set of subobjects of the terminal object of the category of sheaves is canonically a poset, and in fact it is isomorphic to the poset of opens of your given space. Hence you can recover the poset of opens. This is enough to, for instance, detect compactness and locality. In general, you'll need the sobrity condition Qiaochu already mentioned.
$endgroup$
– Ingo Blechschmidt
Dec 14 '18 at 22:43
add a comment |
$begingroup$
At best you can hope to recover the lattice of open subsets, which I think you can do. This recovers the original space if the space is sober but not in general.
$endgroup$
– Qiaochu Yuan
May 26 '18 at 17:15
$begingroup$
A presheaf is a contravariant functor from $O(X)$ to what category? Or what else do you mean by the category of presheaves?
$endgroup$
– Henno Brandsma
May 27 '18 at 6:54
$begingroup$
the set of all open subsets of a topological space is a poset, hence form a category, a functor from this category to the category of sets is a presheaf but it could be any category as the target. I just wanted to know how to express these classical properties of a topological space but now expressed in categorial formalism
$endgroup$
– meta
May 27 '18 at 23:04
1
$begingroup$
by category of presheaves I mean a category where the objects are functors from the category formed by all open subsets to another category
$endgroup$
– meta
May 28 '18 at 3:54
$begingroup$
If you were referring to the category of sheaves instead of the category of presheaves, it would work as follows: The set of subobjects of the terminal object of the category of sheaves is canonically a poset, and in fact it is isomorphic to the poset of opens of your given space. Hence you can recover the poset of opens. This is enough to, for instance, detect compactness and locality. In general, you'll need the sobrity condition Qiaochu already mentioned.
$endgroup$
– Ingo Blechschmidt
Dec 14 '18 at 22:43
$begingroup$
At best you can hope to recover the lattice of open subsets, which I think you can do. This recovers the original space if the space is sober but not in general.
$endgroup$
– Qiaochu Yuan
May 26 '18 at 17:15
$begingroup$
At best you can hope to recover the lattice of open subsets, which I think you can do. This recovers the original space if the space is sober but not in general.
$endgroup$
– Qiaochu Yuan
May 26 '18 at 17:15
$begingroup$
A presheaf is a contravariant functor from $O(X)$ to what category? Or what else do you mean by the category of presheaves?
$endgroup$
– Henno Brandsma
May 27 '18 at 6:54
$begingroup$
A presheaf is a contravariant functor from $O(X)$ to what category? Or what else do you mean by the category of presheaves?
$endgroup$
– Henno Brandsma
May 27 '18 at 6:54
$begingroup$
the set of all open subsets of a topological space is a poset, hence form a category, a functor from this category to the category of sets is a presheaf but it could be any category as the target. I just wanted to know how to express these classical properties of a topological space but now expressed in categorial formalism
$endgroup$
– meta
May 27 '18 at 23:04
$begingroup$
the set of all open subsets of a topological space is a poset, hence form a category, a functor from this category to the category of sets is a presheaf but it could be any category as the target. I just wanted to know how to express these classical properties of a topological space but now expressed in categorial formalism
$endgroup$
– meta
May 27 '18 at 23:04
1
1
$begingroup$
by category of presheaves I mean a category where the objects are functors from the category formed by all open subsets to another category
$endgroup$
– meta
May 28 '18 at 3:54
$begingroup$
by category of presheaves I mean a category where the objects are functors from the category formed by all open subsets to another category
$endgroup$
– meta
May 28 '18 at 3:54
$begingroup$
If you were referring to the category of sheaves instead of the category of presheaves, it would work as follows: The set of subobjects of the terminal object of the category of sheaves is canonically a poset, and in fact it is isomorphic to the poset of opens of your given space. Hence you can recover the poset of opens. This is enough to, for instance, detect compactness and locality. In general, you'll need the sobrity condition Qiaochu already mentioned.
$endgroup$
– Ingo Blechschmidt
Dec 14 '18 at 22:43
$begingroup$
If you were referring to the category of sheaves instead of the category of presheaves, it would work as follows: The set of subobjects of the terminal object of the category of sheaves is canonically a poset, and in fact it is isomorphic to the poset of opens of your given space. Hence you can recover the poset of opens. This is enough to, for instance, detect compactness and locality. In general, you'll need the sobrity condition Qiaochu already mentioned.
$endgroup$
– Ingo Blechschmidt
Dec 14 '18 at 22:43
add a comment |
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$begingroup$
At best you can hope to recover the lattice of open subsets, which I think you can do. This recovers the original space if the space is sober but not in general.
$endgroup$
– Qiaochu Yuan
May 26 '18 at 17:15
$begingroup$
A presheaf is a contravariant functor from $O(X)$ to what category? Or what else do you mean by the category of presheaves?
$endgroup$
– Henno Brandsma
May 27 '18 at 6:54
$begingroup$
the set of all open subsets of a topological space is a poset, hence form a category, a functor from this category to the category of sets is a presheaf but it could be any category as the target. I just wanted to know how to express these classical properties of a topological space but now expressed in categorial formalism
$endgroup$
– meta
May 27 '18 at 23:04
1
$begingroup$
by category of presheaves I mean a category where the objects are functors from the category formed by all open subsets to another category
$endgroup$
– meta
May 28 '18 at 3:54
$begingroup$
If you were referring to the category of sheaves instead of the category of presheaves, it would work as follows: The set of subobjects of the terminal object of the category of sheaves is canonically a poset, and in fact it is isomorphic to the poset of opens of your given space. Hence you can recover the poset of opens. This is enough to, for instance, detect compactness and locality. In general, you'll need the sobrity condition Qiaochu already mentioned.
$endgroup$
– Ingo Blechschmidt
Dec 14 '18 at 22:43