Is every set a filtered colimit of finite sets?












2












$begingroup$


Is the following statement correct in the category of sets?




Let $X$ be any set. Then there exists a filtered small category $I$ and a functor $F:Ito mathrm{Set}$ such that for all $iin I$ the set $F(i)$ is finite, and such that
$$
X ; = ; mathrm{colim}_{iin I} F(i) .
$$




Are there references on results of this type in the literature?






reference-request category-theory limits-colimits







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    One way to generalize this is the notion of a locally finitely presentable category.
    $endgroup$
    – Derek Elkins
    2 days ago
















2












$begingroup$


Is the following statement correct in the category of sets?




Let $X$ be any set. Then there exists a filtered small category $I$ and a functor $F:Ito mathrm{Set}$ such that for all $iin I$ the set $F(i)$ is finite, and such that
$$
X ; = ; mathrm{colim}_{iin I} F(i) .
$$




Are there references on results of this type in the literature?






reference-request category-theory limits-colimits







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    One way to generalize this is the notion of a locally finitely presentable category.
    $endgroup$
    – Derek Elkins
    2 days ago














2












2








2





$begingroup$


Is the following statement correct in the category of sets?




Let $X$ be any set. Then there exists a filtered small category $I$ and a functor $F:Ito mathrm{Set}$ such that for all $iin I$ the set $F(i)$ is finite, and such that
$$
X ; = ; mathrm{colim}_{iin I} F(i) .
$$




Are there references on results of this type in the literature?






reference-request category-theory limits-colimits







share|cite|improve this question











$endgroup$




Is the following statement correct in the category of sets?




Let $X$ be any set. Then there exists a filtered small category $I$ and a functor $F:Ito mathrm{Set}$ such that for all $iin I$ the set $F(i)$ is finite, and such that
$$
X ; = ; mathrm{colim}_{iin I} F(i) .
$$




Are there references on results of this type in the literature?






reference-request category-theory limits-colimits




reference-request category-theory limits-colimits






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 days ago









Andrés E. Caicedo

65.9k8160252




65.9k8160252










asked 2 days ago









geodudegeodude

4,1911344




4,1911344








  • 1




    $begingroup$
    One way to generalize this is the notion of a locally finitely presentable category.
    $endgroup$
    – Derek Elkins
    2 days ago














  • 1




    $begingroup$
    One way to generalize this is the notion of a locally finitely presentable category.
    $endgroup$
    – Derek Elkins
    2 days ago








1




1




$begingroup$
One way to generalize this is the notion of a locally finitely presentable category.
$endgroup$
– Derek Elkins
2 days ago




$begingroup$
One way to generalize this is the notion of a locally finitely presentable category.
$endgroup$
– Derek Elkins
2 days ago










2 Answers
2






active

oldest

votes


















13












$begingroup$

The answer is yes: every set is the union of its finite subsets.



So take $I = P_{text{finite}}(X)$ with as morphisms the inclusion maps, and $F : I to text{Set}$ the inclusion.






share|cite|improve this answer









$endgroup$





















    9












    $begingroup$

    One answer already mentions the diagram of finite subsets of $X$. You would have to check that taking the union of this system actually is the colimit (which is an easy exercise).



    Since you asked for a reference, Locally Presentable and Accessible Categories by J. Adámek and J. Rosický is a great book on this kind of stuff. In particular example 1.2(1) already mentions the diagram of finite subsets.






    share|cite|improve this answer








    New contributor




    Mark Kamsma is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      13












      $begingroup$

      The answer is yes: every set is the union of its finite subsets.



      So take $I = P_{text{finite}}(X)$ with as morphisms the inclusion maps, and $F : I to text{Set}$ the inclusion.






      share|cite|improve this answer









      $endgroup$


















        13












        $begingroup$

        The answer is yes: every set is the union of its finite subsets.



        So take $I = P_{text{finite}}(X)$ with as morphisms the inclusion maps, and $F : I to text{Set}$ the inclusion.






        share|cite|improve this answer









        $endgroup$
















          13












          13








          13





          $begingroup$

          The answer is yes: every set is the union of its finite subsets.



          So take $I = P_{text{finite}}(X)$ with as morphisms the inclusion maps, and $F : I to text{Set}$ the inclusion.






          share|cite|improve this answer









          $endgroup$



          The answer is yes: every set is the union of its finite subsets.



          So take $I = P_{text{finite}}(X)$ with as morphisms the inclusion maps, and $F : I to text{Set}$ the inclusion.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 2 days ago









          rabotarabota

          14.6k32885




          14.6k32885























              9












              $begingroup$

              One answer already mentions the diagram of finite subsets of $X$. You would have to check that taking the union of this system actually is the colimit (which is an easy exercise).



              Since you asked for a reference, Locally Presentable and Accessible Categories by J. Adámek and J. Rosický is a great book on this kind of stuff. In particular example 1.2(1) already mentions the diagram of finite subsets.






              share|cite|improve this answer








              New contributor




              Mark Kamsma is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
              Check out our Code of Conduct.






              $endgroup$


















                9












                $begingroup$

                One answer already mentions the diagram of finite subsets of $X$. You would have to check that taking the union of this system actually is the colimit (which is an easy exercise).



                Since you asked for a reference, Locally Presentable and Accessible Categories by J. Adámek and J. Rosický is a great book on this kind of stuff. In particular example 1.2(1) already mentions the diagram of finite subsets.






                share|cite|improve this answer








                New contributor




                Mark Kamsma is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                Check out our Code of Conduct.






                $endgroup$
















                  9












                  9








                  9





                  $begingroup$

                  One answer already mentions the diagram of finite subsets of $X$. You would have to check that taking the union of this system actually is the colimit (which is an easy exercise).



                  Since you asked for a reference, Locally Presentable and Accessible Categories by J. Adámek and J. Rosický is a great book on this kind of stuff. In particular example 1.2(1) already mentions the diagram of finite subsets.






                  share|cite|improve this answer








                  New contributor




                  Mark Kamsma is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.






                  $endgroup$



                  One answer already mentions the diagram of finite subsets of $X$. You would have to check that taking the union of this system actually is the colimit (which is an easy exercise).



                  Since you asked for a reference, Locally Presentable and Accessible Categories by J. Adámek and J. Rosický is a great book on this kind of stuff. In particular example 1.2(1) already mentions the diagram of finite subsets.







                  share|cite|improve this answer








                  New contributor




                  Mark Kamsma is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.









                  share|cite|improve this answer



                  share|cite|improve this answer






                  New contributor




                  Mark Kamsma is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.









                  answered 2 days ago









                  Mark KamsmaMark Kamsma

                  3065




                  3065




                  New contributor




                  Mark Kamsma is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.





                  New contributor





                  Mark Kamsma is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.






                  Mark Kamsma is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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