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?










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    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?










    share|cite|improve this question
























      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?










      share|cite|improve this question













      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?







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      asked Nov 22 at 17:29









      Karlm

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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 $Hom_C(X,Y)=Hom_D(X,Y)$.






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          • 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











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          up vote
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          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 $Hom_C(X,Y)=Hom_D(X,Y)$.






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          • 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















          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 $Hom_C(X,Y)=Hom_D(X,Y)$.






          share|cite|improve this answer

















          • 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













          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 $Hom_C(X,Y)=Hom_D(X,Y)$.






          share|cite|improve this answer












          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 $Hom_C(X,Y)=Hom_D(X,Y)$.







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          share|cite|improve this answer










          answered Nov 22 at 17:33









          Tsemo Aristide

          55.2k11444




          55.2k11444








          • 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




            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


















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