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 $mathbfSet$
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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 $mathbfSet$
$begingroup$
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 9 hours ago
Andrés E. Caicedo
65.9k8160252
asked 16 hours 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 9 hours ago
Andrés E. Caicedo
65.9k8160252
asked 16 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
15 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
edited 9 hours ago
Andrés E. Caicedo
65.9k8160252
asked 16 hours 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 9 hours ago
Andrés E. Caicedo
65.9k8160252
asked 16 hours ago
geodudegeodude
4,1911344
edited 9 hours ago
Andrés E. Caicedo
65.9k8160252
asked 16 hours ago
geodudegeodude
4,1911344
edited 9 hours ago
Andrés E. Caicedo
65.9k8160252
edited 9 hours ago
Andrés E. Caicedo
65.9k8160252
edited 9 hours ago
Andrés E. Caicedo
65.9k8160252
65.9k8160252
asked 16 hours ago
geodudegeodude
4,1911344
asked 16 hours ago
geodudegeodude
4,1911344
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
add a comment |
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
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 16 hours ago
rabotarabota
14.5k32885
$endgroup$
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 15 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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2 Answers
2
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2 Answers
2
active
oldest
votes
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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 16 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.
answered 16 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.
answered 16 hours ago
rabotarabota
14.5k32885
$endgroup$
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 16 hours ago
rabotarabota
14.5k32885
answered 16 hours ago
rabotarabota
14.5k32885
answered 16 hours ago
rabotarabota
14.5k32885
answered 16 hours ago
rabotarabota
14.5k32885
14.5k32885
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 15 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.
answered 15 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.
answered 15 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.
$endgroup$
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 15 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 15 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 15 hours ago
Mark KamsmaMark Kamsma
1564
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.
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
15 hours ago