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Is every set a filtered colimit of finite sets?


On colim $Hom_A-alg(B, C_i)$Why is the colimit over this filtered index category the object $F(i_0)$?A filtered poset and a filtered diagram (category)The colimit of all finite-dimensional vector spacesWhy do finite limits commute with filtered colimits in the category of abelian groups?Colimit of collection of finite setsExpressing Representation of a Colimit as a LimitFiltered vs Directed colimitsNot-quite-preservation of not-quite-filtered colimitsAbout a specific step in a proof of the fact that filtered colimits and finite limits commute in $mathbfSet$













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 $F:Ito mathrmSet$ such that for all $iin I$ the set $F(i)$ is finite, and such that
$$
X ; = ; mathrmcolim_iin I F(i) .
$$




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
    15 hours 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 $F:Ito mathrmSet$ such that for all $iin I$ the set $F(i)$ is finite, and such that
$$
X ; = ; mathrmcolim_iin I F(i) .
$$




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
    15 hours 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 $F:Ito mathrmSet$ such that for all $iin I$ the set $F(i)$ is finite, and such that
$$
X ; = ; mathrmcolim_iin I F(i) .
$$




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 $F:Ito mathrmSet$ such that for all $iin I$ the set $F(i)$ is finite, and such that
$$
X ; = ; mathrmcolim_iin I F(i) .
$$




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 9 hours ago









Andrés E. Caicedo

65.9k8160252




65.9k8160252










asked 16 hours 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
    15 hours ago












  • 1




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







1




1




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




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










2 Answers
2






active

oldest

votes


















13












$begingroup$

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



So take $I = P_textfinite(X)$ with as morphisms the inclusion maps, and $F : I to textSet$ 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.
    Check out our Code of Conduct.






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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 $I = P_textfinite(X)$ with as morphisms the inclusion maps, and $F : I to textSet$ 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 $I = P_textfinite(X)$ with as morphisms the inclusion maps, and $F : I to textSet$ 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 $I = P_textfinite(X)$ with as morphisms the inclusion maps, and $F : I to textSet$ the inclusion.






          share|cite|improve this answer









          $endgroup$



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



          So take $I = P_textfinite(X)$ with as morphisms the inclusion maps, and $F : I to textSet$ the inclusion.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 16 hours ago









          rabotarabota

          14.5k32885




          14.5k32885





















              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 15 hours ago









                  Mark KamsmaMark Kamsma

                  1564




                  1564




                  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.
                  Check out our Code of Conduct.



























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