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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

edited 3 hours ago

Andrés E. Caicedo

65.9k8160252

asked 9 hours ago

geodudegeodude

4,1911344

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

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

  
  
  
  

add a comment | 

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

edited 3 hours ago

Andrés E. Caicedo

65.9k8160252

asked 9 hours ago

geodudegeodude

4,1911344

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

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

  
  
  
  

add a comment | 

$begingroup$

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

edited 3 hours ago

Andrés E. Caicedo

65.9k8160252

asked 9 hours ago

geodudegeodude

4,1911344

$endgroup$

share|cite|improve this question

edited 3 hours ago

Andrés E. Caicedo

65.9k8160252

asked 9 hours ago

geodudegeodude

4,1911344

share|cite|improve this question

edited 3 hours ago

Andrés E. Caicedo

65.9k8160252

asked 9 hours ago

geodudegeodude

4,1911344

edited 3 hours ago

Andrés E. Caicedo

65.9k8160252

edited 3 hours ago

Andrés E. Caicedo

65.9k8160252

asked 9 hours ago

geodudegeodude

4,1911344

asked 9 hours ago

geodudegeodude

4,1911344

2 Answers
2

active

oldest

votes

12

$begingroup$

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

answered 9 hours ago

rabotarabota

14.5k32885

$endgroup$

add a comment | 

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

answered 9 hours ago

Mark KamsmaMark Kamsma

1363

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.

$endgroup$

add a comment | 

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12

$begingroup$

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

answered 9 hours ago

rabotarabota

14.5k32885

$endgroup$

add a comment | 

12

$begingroup$

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

answered 9 hours ago

rabotarabota

14.5k32885

$endgroup$

add a comment | 

$begingroup$

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

answered 9 hours ago

rabotarabota

14.5k32885

$endgroup$

share|cite|improve this answer

answered 9 hours ago

rabotarabota

14.5k32885

answered 9 hours ago

rabotarabota

14.5k32885

answered 9 hours ago

rabotarabota

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

answered 9 hours ago

Mark KamsmaMark Kamsma

1363

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.

$endgroup$

add a comment | 

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

answered 9 hours ago

Mark KamsmaMark Kamsma

1363

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.

$endgroup$

add a comment | 

$begingroup$

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

answered 9 hours ago

Mark KamsmaMark Kamsma

1363

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.

$endgroup$

share|cite|improve this answer

answered 9 hours ago

Mark KamsmaMark Kamsma

1363

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 9 hours ago

Mark KamsmaMark Kamsma

1363

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 9 hours ago

Mark KamsmaMark Kamsma

1363