Is every set a filtered colimit of finite sets? The 2019 Stack Overflow Developer Survey Results Are InOn 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 $mathbfSet$
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Is every set a filtered colimit of finite sets?
The 2019 Stack Overflow Developer Survey Results Are InOn 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 $mathbfSet$
$begingroup$
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
edited 2 days ago
Andrés E. Caicedo
65.9k8160252
asked 2 days ago
geodudegeodude
4,1911344
$endgroup$
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
edited 2 days ago
Andrés E. Caicedo
65.9k8160252
asked 2 days ago
geodudegeodude
4,1911344
$endgroup$
1
$begingroup$
One way to generalize this is the notion of a locally finitely presentable category.
$endgroup$
– Derek Elkins
2 days 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
edited 2 days ago
Andrés E. Caicedo
65.9k8160252
asked 2 days ago
geodudegeodude
4,1911344
$endgroup$
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
edited 2 days ago
Andrés E. Caicedo
65.9k8160252
asked 2 days ago
geodudegeodude
4,1911344
edited 2 days ago
Andrés E. Caicedo
65.9k8160252
asked 2 days ago
geodudegeodude
4,1911344
edited 2 days ago
Andrés E. Caicedo
65.9k8160252
edited 2 days ago
Andrés E. Caicedo
65.9k8160252
edited 2 days ago
Andrés E. Caicedo
65.9k8160252
65.9k8160252
asked 2 days ago
geodudegeodude
4,1911344
asked 2 days ago
geodudegeodude
4,1911344
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
add a comment |
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
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
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.
answered 2 days ago
rabotarabota
14.6k32885
$endgroup$
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.
answered 2 days ago
Mark KamsmaMark Kamsma
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.
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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.
answered 2 days ago
rabotarabota
14.6k32885
$endgroup$
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.
answered 2 days ago
rabotarabota
14.6k32885
$endgroup$
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.
answered 2 days ago
rabotarabota
14.6k32885
$endgroup$
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.
answered 2 days ago
rabotarabota
14.6k32885
answered 2 days ago
rabotarabota
14.6k32885
answered 2 days ago
rabotarabota
14.6k32885
answered 2 days ago
rabotarabota
14.6k32885
14.6k32885
add a comment |
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.
answered 2 days ago
Mark KamsmaMark Kamsma
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.
$endgroup$
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.
answered 2 days ago
Mark KamsmaMark Kamsma
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.
$endgroup$
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.
answered 2 days ago
Mark KamsmaMark Kamsma
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.
$endgroup$
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.
answered 2 days ago
Mark KamsmaMark Kamsma
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.
answered 2 days ago
Mark KamsmaMark Kamsma
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.
answered 2 days ago
Mark KamsmaMark Kamsma
3065
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.
Check out our Code of Conduct.
add a comment |
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$begingroup$
One way to generalize this is the notion of a locally finitely presentable category.
$endgroup$
– Derek Elkins
2 days ago