Mixing
Can we deduce that morphisms in categories of structures should be “structure preserving”
$begingroup$
One of the fundamental ways category theory is used is to define categories of structures, where the morphisms are structure preserving maps. E.g. the category of topologies has continuous functions as morphisms.
But as far as I know, this is just a customary way that category theory can be used. We could just as well have said that the category of topologies has all functions as morphisms including non-continuous ones.
This would of course be useless, but apart from its practical use, is there also some sort of principled deduction that “structure preserving maps as morphisms” is somehow “the right way” or “a canonical way” of defining categories?
category-theory
asked Jan 31 at 11:17
user56834user56834
3,42321253
$endgroup$
add a comment |
$begingroup$
One of the fundamental ways category theory is used is to define categories of structures, where the morphisms are structure preserving maps. E.g. the category of topologies has continuous functions as morphisms.
But as far as I know, this is just a customary way that category theory can be used. We could just as well have said that the category of topologies has all functions as morphisms including non-continuous ones.
This would of course be useless, but apart from its practical use, is there also some sort of principled deduction that “structure preserving maps as morphisms” is somehow “the right way” or “a canonical way” of defining categories?
category-theory
asked Jan 31 at 11:17
user56834user56834
3,42321253
$endgroup$
$begingroup$
It's just that we, mathematicians, are not interested in noncontinuous maps between topological spaces (most of the time); so what would be the point of considering this category ? Similarly if I called $X$ the set of subsets $F$ of $mathbb{R}$ such that if there exists a positive $xin F$, then $1in F$; then I can define $X$, but who cares ?
$endgroup$
– Max
Jan 31 at 12:23
$begingroup$
@max, you are attacking straw man alternatives to the way it’s generally done. My question is motivated by the fact that I find it interesting that there is this general concept of “structure preserving maps” that are always used in category theory. This makes me wonder whether there is a deeper reason why structure preserving is a fundamental concept.
$endgroup$
– user56834
Jan 31 at 12:32
$begingroup$
Structure preserving is a fundamental concept because we're interested in structures, there's not much more than that.
$endgroup$
– Max
Jan 31 at 12:35
$begingroup$
If we were to deduce this, whatever we mean by that, we'd be saying that the categories that aren't structures with structure preserving maps are pathological. But very, very many categories that are not of this sort are actually interesting and useful.
$endgroup$
– Malice Vidrine
Feb 2 at 22:08
add a comment |
$begingroup$
One of the fundamental ways category theory is used is to define categories of structures, where the morphisms are structure preserving maps. E.g. the category of topologies has continuous functions as morphisms.
But as far as I know, this is just a customary way that category theory can be used. We could just as well have said that the category of topologies has all functions as morphisms including non-continuous ones.
This would of course be useless, but apart from its practical use, is there also some sort of principled deduction that “structure preserving maps as morphisms” is somehow “the right way” or “a canonical way” of defining categories?
category-theory
asked Jan 31 at 11:17
user56834user56834
3,42321253
$endgroup$
One of the fundamental ways category theory is used is to define categories of structures, where the morphisms are structure preserving maps. E.g. the category of topologies has continuous functions as morphisms.
But as far as I know, this is just a customary way that category theory can be used. We could just as well have said that the category of topologies has all functions as morphisms including non-continuous ones.
This would of course be useless, but apart from its practical use, is there also some sort of principled deduction that “structure preserving maps as morphisms” is somehow “the right way” or “a canonical way” of defining categories?
category-theory
category-theory
asked Jan 31 at 11:17
user56834user56834
3,42321253
asked Jan 31 at 11:17
user56834user56834
3,42321253
asked Jan 31 at 11:17
user56834user56834
3,42321253
asked Jan 31 at 11:17
user56834user56834
3,42321253
asked Jan 31 at 11:17
user56834user56834
3,42321253
3,42321253
$begingroup$
It's just that we, mathematicians, are not interested in noncontinuous maps between topological spaces (most of the time); so what would be the point of considering this category ? Similarly if I called $X$ the set of subsets $F$ of $mathbb{R}$ such that if there exists a positive $xin F$, then $1in F$; then I can define $X$, but who cares ?
$endgroup$
– Max
Jan 31 at 12:23
$begingroup$
@max, you are attacking straw man alternatives to the way it’s generally done. My question is motivated by the fact that I find it interesting that there is this general concept of “structure preserving maps” that are always used in category theory. This makes me wonder whether there is a deeper reason why structure preserving is a fundamental concept.
$endgroup$
– user56834
Jan 31 at 12:32
$begingroup$
Structure preserving is a fundamental concept because we're interested in structures, there's not much more than that.
$endgroup$
– Max
Jan 31 at 12:35
$begingroup$
If we were to deduce this, whatever we mean by that, we'd be saying that the categories that aren't structures with structure preserving maps are pathological. But very, very many categories that are not of this sort are actually interesting and useful.
$endgroup$
– Malice Vidrine
Feb 2 at 22:08
add a comment |
$begingroup$
It's just that we, mathematicians, are not interested in noncontinuous maps between topological spaces (most of the time); so what would be the point of considering this category ? Similarly if I called $X$ the set of subsets $F$ of $mathbb{R}$ such that if there exists a positive $xin F$, then $1in F$; then I can define $X$, but who cares ?
$endgroup$
– Max
Jan 31 at 12:23
$begingroup$
@max, you are attacking straw man alternatives to the way it’s generally done. My question is motivated by the fact that I find it interesting that there is this general concept of “structure preserving maps” that are always used in category theory. This makes me wonder whether there is a deeper reason why structure preserving is a fundamental concept.
$endgroup$
– user56834
Jan 31 at 12:32
$begingroup$
Structure preserving is a fundamental concept because we're interested in structures, there's not much more than that.
$endgroup$
– Max
Jan 31 at 12:35
$begingroup$
If we were to deduce this, whatever we mean by that, we'd be saying that the categories that aren't structures with structure preserving maps are pathological. But very, very many categories that are not of this sort are actually interesting and useful.
$endgroup$
– Malice Vidrine
Feb 2 at 22:08
$begingroup$
It's just that we, mathematicians, are not interested in noncontinuous maps between topological spaces (most of the time); so what would be the point of considering this category ? Similarly if I called $X$ the set of subsets $F$ of $mathbb{R}$ such that if there exists a positive $xin F$, then $1in F$; then I can define $X$, but who cares ?
$endgroup$
– Max
Jan 31 at 12:23
$begingroup$
It's just that we, mathematicians, are not interested in noncontinuous maps between topological spaces (most of the time); so what would be the point of considering this category ? Similarly if I called $X$ the set of subsets $F$ of $mathbb{R}$ such that if there exists a positive $xin F$, then $1in F$; then I can define $X$, but who cares ?
$endgroup$
– Max
Jan 31 at 12:23
$begingroup$
@max, you are attacking straw man alternatives to the way it’s generally done. My question is motivated by the fact that I find it interesting that there is this general concept of “structure preserving maps” that are always used in category theory. This makes me wonder whether there is a deeper reason why structure preserving is a fundamental concept.
$endgroup$
– user56834
Jan 31 at 12:32
$begingroup$
@max, you are attacking straw man alternatives to the way it’s generally done. My question is motivated by the fact that I find it interesting that there is this general concept of “structure preserving maps” that are always used in category theory. This makes me wonder whether there is a deeper reason why structure preserving is a fundamental concept.
$endgroup$
– user56834
Jan 31 at 12:32
$begingroup$
Structure preserving is a fundamental concept because we're interested in structures, there's not much more than that.
$endgroup$
– Max
Jan 31 at 12:35
$begingroup$
Structure preserving is a fundamental concept because we're interested in structures, there's not much more than that.
$endgroup$
– Max
Jan 31 at 12:35
$begingroup$
If we were to deduce this, whatever we mean by that, we'd be saying that the categories that aren't structures with structure preserving maps are pathological. But very, very many categories that are not of this sort are actually interesting and useful.
$endgroup$
– Malice Vidrine
Feb 2 at 22:08
$begingroup$
If we were to deduce this, whatever we mean by that, we'd be saying that the categories that aren't structures with structure preserving maps are pathological. But very, very many categories that are not of this sort are actually interesting and useful.
$endgroup$
– Malice Vidrine
Feb 2 at 22:08
add a comment |
2 Answers
2
active
oldest
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$begingroup$
Category theory doesn't really care about the structure of its objects and morphisms. For instance, the category of topological spaces is made up of all topological spaces and all continuous functions between them and at that point the category theory stops caring about what the objects and the arrows are. Just whether they exist, whether certain compositions are equal and so on.
answered Jan 31 at 11:27
ArthurArthur
122k7122211
$endgroup$
$begingroup$
I know, but there is a reason that these categories are chosen and not others. I’m wondering whether there is something behind that practical reason.
$endgroup$
– user56834
Jan 31 at 11:39
$begingroup$
@user56834 Even though the category itself doesn't explicitly carry any information about open sets or elements, suprisingly much of that information is encoded in the category structure. For instance, any topological space has exactly one morphism to any singleton space, and only singleton spaces have this property ("terminal object" is the categorical term). Once you have identified the singleton spaces, you can see constant maps as maps which "factor through" a singleton space. Many other things like injectivity or homeomorphicity are also encoded in similar fashion.
$endgroup$
– Arthur
Jan 31 at 12:07
add a comment |
$begingroup$
There are two things to keep in mind.
First, as set theory treats sets as structure-less collections, i.e. lists of unnamed elements (it does not care if its elements are real numbers, apples, people), category theoretic objects have no internal structure.
The point of category theory is exactly to study structures without looking to their internal structure, but doing so makes impossible to deal with the structure preserving-property.
Another point of view is that the morphisms give the structure to the objects. There is also a very technical way to make this formal, you can find more in this answer.
Hope this helps.
answered Feb 2 at 16:39
Giorgio MossaGiorgio Mossa
14.3k11749
$endgroup$
add a comment |
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2 Answers
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2 Answers
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$begingroup$
Category theory doesn't really care about the structure of its objects and morphisms. For instance, the category of topological spaces is made up of all topological spaces and all continuous functions between them and at that point the category theory stops caring about what the objects and the arrows are. Just whether they exist, whether certain compositions are equal and so on.
answered Jan 31 at 11:27
ArthurArthur
122k7122211
$endgroup$
$begingroup$
I know, but there is a reason that these categories are chosen and not others. I’m wondering whether there is something behind that practical reason.
$endgroup$
– user56834
Jan 31 at 11:39
$begingroup$
@user56834 Even though the category itself doesn't explicitly carry any information about open sets or elements, suprisingly much of that information is encoded in the category structure. For instance, any topological space has exactly one morphism to any singleton space, and only singleton spaces have this property ("terminal object" is the categorical term). Once you have identified the singleton spaces, you can see constant maps as maps which "factor through" a singleton space. Many other things like injectivity or homeomorphicity are also encoded in similar fashion.
$endgroup$
– Arthur
Jan 31 at 12:07
add a comment |
$begingroup$
Category theory doesn't really care about the structure of its objects and morphisms. For instance, the category of topological spaces is made up of all topological spaces and all continuous functions between them and at that point the category theory stops caring about what the objects and the arrows are. Just whether they exist, whether certain compositions are equal and so on.
answered Jan 31 at 11:27
ArthurArthur
122k7122211
$endgroup$
$begingroup$
I know, but there is a reason that these categories are chosen and not others. I’m wondering whether there is something behind that practical reason.
$endgroup$
– user56834
Jan 31 at 11:39
$begingroup$
@user56834 Even though the category itself doesn't explicitly carry any information about open sets or elements, suprisingly much of that information is encoded in the category structure. For instance, any topological space has exactly one morphism to any singleton space, and only singleton spaces have this property ("terminal object" is the categorical term). Once you have identified the singleton spaces, you can see constant maps as maps which "factor through" a singleton space. Many other things like injectivity or homeomorphicity are also encoded in similar fashion.
$endgroup$
– Arthur
Jan 31 at 12:07
add a comment |
$begingroup$
Category theory doesn't really care about the structure of its objects and morphisms. For instance, the category of topological spaces is made up of all topological spaces and all continuous functions between them and at that point the category theory stops caring about what the objects and the arrows are. Just whether they exist, whether certain compositions are equal and so on.
answered Jan 31 at 11:27
ArthurArthur
122k7122211
$endgroup$
Category theory doesn't really care about the structure of its objects and morphisms. For instance, the category of topological spaces is made up of all topological spaces and all continuous functions between them and at that point the category theory stops caring about what the objects and the arrows are. Just whether they exist, whether certain compositions are equal and so on.
answered Jan 31 at 11:27
ArthurArthur
122k7122211
answered Jan 31 at 11:27
ArthurArthur
122k7122211
answered Jan 31 at 11:27
ArthurArthur
122k7122211
answered Jan 31 at 11:27
ArthurArthur
122k7122211
122k7122211
$begingroup$
I know, but there is a reason that these categories are chosen and not others. I’m wondering whether there is something behind that practical reason.
$endgroup$
– user56834
Jan 31 at 11:39
$begingroup$
@user56834 Even though the category itself doesn't explicitly carry any information about open sets or elements, suprisingly much of that information is encoded in the category structure. For instance, any topological space has exactly one morphism to any singleton space, and only singleton spaces have this property ("terminal object" is the categorical term). Once you have identified the singleton spaces, you can see constant maps as maps which "factor through" a singleton space. Many other things like injectivity or homeomorphicity are also encoded in similar fashion.
$endgroup$
– Arthur
Jan 31 at 12:07
add a comment |
$begingroup$
I know, but there is a reason that these categories are chosen and not others. I’m wondering whether there is something behind that practical reason.
$endgroup$
– user56834
Jan 31 at 11:39
$begingroup$
@user56834 Even though the category itself doesn't explicitly carry any information about open sets or elements, suprisingly much of that information is encoded in the category structure. For instance, any topological space has exactly one morphism to any singleton space, and only singleton spaces have this property ("terminal object" is the categorical term). Once you have identified the singleton spaces, you can see constant maps as maps which "factor through" a singleton space. Many other things like injectivity or homeomorphicity are also encoded in similar fashion.
$endgroup$
– Arthur
Jan 31 at 12:07
$begingroup$
I know, but there is a reason that these categories are chosen and not others. I’m wondering whether there is something behind that practical reason.
$endgroup$
– user56834
Jan 31 at 11:39
$begingroup$
I know, but there is a reason that these categories are chosen and not others. I’m wondering whether there is something behind that practical reason.
$endgroup$
– user56834
Jan 31 at 11:39
$begingroup$
@user56834 Even though the category itself doesn't explicitly carry any information about open sets or elements, suprisingly much of that information is encoded in the category structure. For instance, any topological space has exactly one morphism to any singleton space, and only singleton spaces have this property ("terminal object" is the categorical term). Once you have identified the singleton spaces, you can see constant maps as maps which "factor through" a singleton space. Many other things like injectivity or homeomorphicity are also encoded in similar fashion.
$endgroup$
– Arthur
Jan 31 at 12:07
$begingroup$
@user56834 Even though the category itself doesn't explicitly carry any information about open sets or elements, suprisingly much of that information is encoded in the category structure. For instance, any topological space has exactly one morphism to any singleton space, and only singleton spaces have this property ("terminal object" is the categorical term). Once you have identified the singleton spaces, you can see constant maps as maps which "factor through" a singleton space. Many other things like injectivity or homeomorphicity are also encoded in similar fashion.
$endgroup$
– Arthur
Jan 31 at 12:07
add a comment |
$begingroup$
There are two things to keep in mind.
First, as set theory treats sets as structure-less collections, i.e. lists of unnamed elements (it does not care if its elements are real numbers, apples, people), category theoretic objects have no internal structure.
The point of category theory is exactly to study structures without looking to their internal structure, but doing so makes impossible to deal with the structure preserving-property.
Another point of view is that the morphisms give the structure to the objects. There is also a very technical way to make this formal, you can find more in this answer.
Hope this helps.
answered Feb 2 at 16:39
Giorgio MossaGiorgio Mossa
14.3k11749
$endgroup$
add a comment |
$begingroup$
There are two things to keep in mind.
First, as set theory treats sets as structure-less collections, i.e. lists of unnamed elements (it does not care if its elements are real numbers, apples, people), category theoretic objects have no internal structure.
The point of category theory is exactly to study structures without looking to their internal structure, but doing so makes impossible to deal with the structure preserving-property.
Another point of view is that the morphisms give the structure to the objects. There is also a very technical way to make this formal, you can find more in this answer.
Hope this helps.
answered Feb 2 at 16:39
Giorgio MossaGiorgio Mossa
14.3k11749
$endgroup$
add a comment |
$begingroup$
There are two things to keep in mind.
First, as set theory treats sets as structure-less collections, i.e. lists of unnamed elements (it does not care if its elements are real numbers, apples, people), category theoretic objects have no internal structure.
The point of category theory is exactly to study structures without looking to their internal structure, but doing so makes impossible to deal with the structure preserving-property.
Another point of view is that the morphisms give the structure to the objects. There is also a very technical way to make this formal, you can find more in this answer.
Hope this helps.
answered Feb 2 at 16:39
Giorgio MossaGiorgio Mossa
14.3k11749
$endgroup$
There are two things to keep in mind.
First, as set theory treats sets as structure-less collections, i.e. lists of unnamed elements (it does not care if its elements are real numbers, apples, people), category theoretic objects have no internal structure.
The point of category theory is exactly to study structures without looking to their internal structure, but doing so makes impossible to deal with the structure preserving-property.
Another point of view is that the morphisms give the structure to the objects. There is also a very technical way to make this formal, you can find more in this answer.
Hope this helps.
answered Feb 2 at 16:39
Giorgio MossaGiorgio Mossa
14.3k11749
answered Feb 2 at 16:39
Giorgio MossaGiorgio Mossa
14.3k11749
answered Feb 2 at 16:39
Giorgio MossaGiorgio Mossa
14.3k11749
answered Feb 2 at 16:39
Giorgio MossaGiorgio Mossa
14.3k11749
14.3k11749
add a comment |
add a comment |
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$begingroup$
It's just that we, mathematicians, are not interested in noncontinuous maps between topological spaces (most of the time); so what would be the point of considering this category ? Similarly if I called $X$ the set of subsets $F$ of $mathbb{R}$ such that if there exists a positive $xin F$, then $1in F$; then I can define $X$, but who cares ?
$endgroup$
– Max
Jan 31 at 12:23
$begingroup$
@max, you are attacking straw man alternatives to the way it’s generally done. My question is motivated by the fact that I find it interesting that there is this general concept of “structure preserving maps” that are always used in category theory. This makes me wonder whether there is a deeper reason why structure preserving is a fundamental concept.
$endgroup$
– user56834
Jan 31 at 12:32
$begingroup$
Structure preserving is a fundamental concept because we're interested in structures, there's not much more than that.
$endgroup$
– Max
Jan 31 at 12:35
$begingroup$
If we were to deduce this, whatever we mean by that, we'd be saying that the categories that aren't structures with structure preserving maps are pathological. But very, very many categories that are not of this sort are actually interesting and useful.
$endgroup$
– Malice Vidrine
Feb 2 at 22:08