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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 $mathbf{Set}$

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

edited 10 hours ago

Andrés E. Caicedo

65.9k8160252

asked 17 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
  17 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 $F:Ito mathrm{Set}$ 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

edited 10 hours ago

Andrés E. Caicedo

65.9k8160252

asked 17 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
  17 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 $F:Ito mathrm{Set}$ 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

edited 10 hours ago

Andrés E. Caicedo

65.9k8160252

asked 17 hours ago

geodudegeodude

4,1911344

$endgroup$

share|cite|improve this question

edited 10 hours ago

Andrés E. Caicedo

65.9k8160252

asked 17 hours ago

geodudegeodude

4,1911344

share|cite|improve this question

edited 10 hours ago

Andrés E. Caicedo

65.9k8160252

asked 17 hours ago

geodudegeodude

4,1911344

edited 10 hours ago

Andrés E. Caicedo

65.9k8160252

edited 10 hours ago

Andrés E. Caicedo

65.9k8160252

asked 17 hours ago

geodudegeodude

4,1911344

asked 17 hours ago

geodudegeodude

4,1911344

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

answered 17 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 17 hours ago

Mark KamsmaMark Kamsma

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.

$endgroup$

add a comment | 

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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

answered 17 hours ago

rabotarabota

14.5k32885

$endgroup$

add a comment | 

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

answered 17 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 $I = P_{text{finite}}(X)$ with as morphisms the inclusion maps, and $F : I to text{Set}$ the inclusion.

share|cite|improve this answer

answered 17 hours ago

rabotarabota

14.5k32885

$endgroup$

share|cite|improve this answer

answered 17 hours ago

rabotarabota

14.5k32885

answered 17 hours ago

rabotarabota

14.5k32885

answered 17 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 17 hours ago

Mark KamsmaMark Kamsma

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.

$endgroup$

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 17 hours ago

Mark KamsmaMark Kamsma

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.

$endgroup$

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 17 hours ago

Mark KamsmaMark Kamsma

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.

$endgroup$

share|cite|improve this answer

answered 17 hours ago

Mark KamsmaMark Kamsma

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.

answered 17 hours ago

Mark KamsmaMark Kamsma

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.

answered 17 hours ago

Mark KamsmaMark Kamsma

1564