A category is discrete if and only if all its subcategories are full
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I am trying prove this. It is stated on the wikipedia page about discrete categories.
One way of the proof is very easy and obvious: if a category is discrete then all its subcategories are also discrete and, hence, full (they all have the identities, wich are the only arrows in the first category. So they all have all the arrows relative to the objects in the first category).
The other way is trickier. We consider some category $C$. If all its subcategories are full, then $C$ must be discrete.
I think I can see why this must be so, but I can't state a general proof. I'm thinking this way:
Consider a category $C$ with only two objects, $a$ and $b$, and a single arrow $f: arightarrow b$ between them (besides the identities, of course). Then it has four subcategories, but one of them is not full, i.e., the subcategory that has both objects, but not the $f$ arrow. So it seems the only way for all subcategories of $C$ to be full is $C$ being discrete. Otherwise there will always be a discrete subcategory that is not full.
Is this reasoning correct? How can I make it more general?
category-theory
asked Nov 22 at 17:29
Karlm
505
add a comment |
up vote
2
down vote
favorite
I am trying prove this. It is stated on the wikipedia page about discrete categories.
One way of the proof is very easy and obvious: if a category is discrete then all its subcategories are also discrete and, hence, full (they all have the identities, wich are the only arrows in the first category. So they all have all the arrows relative to the objects in the first category).
The other way is trickier. We consider some category $C$. If all its subcategories are full, then $C$ must be discrete.
I think I can see why this must be so, but I can't state a general proof. I'm thinking this way:
Consider a category $C$ with only two objects, $a$ and $b$, and a single arrow $f: arightarrow b$ between them (besides the identities, of course). Then it has four subcategories, but one of them is not full, i.e., the subcategory that has both objects, but not the $f$ arrow. So it seems the only way for all subcategories of $C$ to be full is $C$ being discrete. Otherwise there will always be a discrete subcategory that is not full.
Is this reasoning correct? How can I make it more general?
category-theory
asked Nov 22 at 17:29
Karlm
505
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I am trying prove this. It is stated on the wikipedia page about discrete categories.
One way of the proof is very easy and obvious: if a category is discrete then all its subcategories are also discrete and, hence, full (they all have the identities, wich are the only arrows in the first category. So they all have all the arrows relative to the objects in the first category).
The other way is trickier. We consider some category $C$. If all its subcategories are full, then $C$ must be discrete.
I think I can see why this must be so, but I can't state a general proof. I'm thinking this way:
Consider a category $C$ with only two objects, $a$ and $b$, and a single arrow $f: arightarrow b$ between them (besides the identities, of course). Then it has four subcategories, but one of them is not full, i.e., the subcategory that has both objects, but not the $f$ arrow. So it seems the only way for all subcategories of $C$ to be full is $C$ being discrete. Otherwise there will always be a discrete subcategory that is not full.
Is this reasoning correct? How can I make it more general?
category-theory
asked Nov 22 at 17:29
Karlm
505
I am trying prove this. It is stated on the wikipedia page about discrete categories.
One way of the proof is very easy and obvious: if a category is discrete then all its subcategories are also discrete and, hence, full (they all have the identities, wich are the only arrows in the first category. So they all have all the arrows relative to the objects in the first category).
The other way is trickier. We consider some category $C$. If all its subcategories are full, then $C$ must be discrete.
I think I can see why this must be so, but I can't state a general proof. I'm thinking this way:
Consider a category $C$ with only two objects, $a$ and $b$, and a single arrow $f: arightarrow b$ between them (besides the identities, of course). Then it has four subcategories, but one of them is not full, i.e., the subcategory that has both objects, but not the $f$ arrow. So it seems the only way for all subcategories of $C$ to be full is $C$ being discrete. Otherwise there will always be a discrete subcategory that is not full.
Is this reasoning correct? How can I make it more general?
category-theory
category-theory
asked Nov 22 at 17:29
Karlm
505
asked Nov 22 at 17:29
Karlm
505
asked Nov 22 at 17:29
Karlm
505
asked Nov 22 at 17:29
Karlm
505
asked Nov 22 at 17:29
Karlm
505
505
add a comment |
add a comment |
1 Answer
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Consider the subcategory $D$ of $C$ which has the same set of objects and $Hom_D(X,Y)$ is empty if $Xneq Y$ and $Hom_D(X,X)={Id_X}$ it is full. So $Hom_C(X,Y)=Hom_D(X,Y)$.
answered Nov 22 at 17:33
Tsemo Aristide
55k11444
1
Ok, let me see if I get this. For any category $C$ we have its discrete subcategory $D$ with same objects but only identities. By our supposition, this category is full. But then this means that $C$ can't have any arrow that is not in $D$, otherwise $D$ would not be full. So $C$ is also discrete (and in fact identical to $D$).
– Karlm
Nov 22 at 17:47
add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
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up vote
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accepted
Consider the subcategory $D$ of $C$ which has the same set of objects and $Hom_D(X,Y)$ is empty if $Xneq Y$ and $Hom_D(X,X)={Id_X}$ it is full. So $Hom_C(X,Y)=Hom_D(X,Y)$.
answered Nov 22 at 17:33
Tsemo Aristide
55k11444
1
Ok, let me see if I get this. For any category $C$ we have its discrete subcategory $D$ with same objects but only identities. By our supposition, this category is full. But then this means that $C$ can't have any arrow that is not in $D$, otherwise $D$ would not be full. So $C$ is also discrete (and in fact identical to $D$).
– Karlm
Nov 22 at 17:47
add a comment |
up vote
4
down vote
accepted
Consider the subcategory $D$ of $C$ which has the same set of objects and $Hom_D(X,Y)$ is empty if $Xneq Y$ and $Hom_D(X,X)={Id_X}$ it is full. So $Hom_C(X,Y)=Hom_D(X,Y)$.
answered Nov 22 at 17:33
Tsemo Aristide
55k11444
1
Ok, let me see if I get this. For any category $C$ we have its discrete subcategory $D$ with same objects but only identities. By our supposition, this category is full. But then this means that $C$ can't have any arrow that is not in $D$, otherwise $D$ would not be full. So $C$ is also discrete (and in fact identical to $D$).
– Karlm
Nov 22 at 17:47
add a comment |
up vote
4
down vote
accepted
up vote
4
down vote
accepted
Consider the subcategory $D$ of $C$ which has the same set of objects and $Hom_D(X,Y)$ is empty if $Xneq Y$ and $Hom_D(X,X)={Id_X}$ it is full. So $Hom_C(X,Y)=Hom_D(X,Y)$.
answered Nov 22 at 17:33
Tsemo Aristide
55k11444
Consider the subcategory $D$ of $C$ which has the same set of objects and $Hom_D(X,Y)$ is empty if $Xneq Y$ and $Hom_D(X,X)={Id_X}$ it is full. So $Hom_C(X,Y)=Hom_D(X,Y)$.
answered Nov 22 at 17:33
Tsemo Aristide
55k11444
answered Nov 22 at 17:33
Tsemo Aristide
55k11444
answered Nov 22 at 17:33
Tsemo Aristide
55k11444
answered Nov 22 at 17:33
Tsemo Aristide
55k11444
55k11444
1
Ok, let me see if I get this. For any category $C$ we have its discrete subcategory $D$ with same objects but only identities. By our supposition, this category is full. But then this means that $C$ can't have any arrow that is not in $D$, otherwise $D$ would not be full. So $C$ is also discrete (and in fact identical to $D$).
– Karlm
Nov 22 at 17:47
add a comment |
1
Ok, let me see if I get this. For any category $C$ we have its discrete subcategory $D$ with same objects but only identities. By our supposition, this category is full. But then this means that $C$ can't have any arrow that is not in $D$, otherwise $D$ would not be full. So $C$ is also discrete (and in fact identical to $D$).
– Karlm
Nov 22 at 17:47
1
1
Ok, let me see if I get this. For any category $C$ we have its discrete subcategory $D$ with same objects but only identities. By our supposition, this category is full. But then this means that $C$ can't have any arrow that is not in $D$, otherwise $D$ would not be full. So $C$ is also discrete (and in fact identical to $D$).
– Karlm
Nov 22 at 17:47
Ok, let me see if I get this. For any category $C$ we have its discrete subcategory $D$ with same objects but only identities. By our supposition, this category is full. But then this means that $C$ can't have any arrow that is not in $D$, otherwise $D$ would not be full. So $C$ is also discrete (and in fact identical to $D$).
– Karlm
Nov 22 at 17:47
add a comment |
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