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Factorization of colimits through slices?
Clash Royale CLAN TAG#URR8PPP
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I could swear I remember a result of the following form:
Suppose we have a pair of functors $$CxrightarrowFDxrightarrowG X,$$ with $X$ cocomplete.
then we obtain a functor $$Dto X$$ sending $$dmapsto operatornamecolim_(Fdownarrow d)(Gcirc pi_d)$$ where $pi_d:(Fdownarrow d)to D$ is the projection functor sending $(F(c)to d) mapsto F(c)$.
Then there is a canonical isomorphism $$operatornamecolim_Dleft (operatornamecolim_(Fdownarrow d)(Gcirc pi_d)right) cong operatornamecolim_C (Gcirc F).$$
Is this true? Do you know of a source? Is there a name for this kind of result?
ct.category-theory
asked 4 hours ago
Harry Gindi
8,336674160
 |Â
show 1 more comment
up vote
3
down vote
favorite
I could swear I remember a result of the following form:
Suppose we have a pair of functors $$CxrightarrowFDxrightarrowG X,$$ with $X$ cocomplete.
then we obtain a functor $$Dto X$$ sending $$dmapsto operatornamecolim_(Fdownarrow d)(Gcirc pi_d)$$ where $pi_d:(Fdownarrow d)to D$ is the projection functor sending $(F(c)to d) mapsto F(c)$.
Then there is a canonical isomorphism $$operatornamecolim_Dleft (operatornamecolim_(Fdownarrow d)(Gcirc pi_d)right) cong operatornamecolim_C (Gcirc F).$$
Is this true? Do you know of a source? Is there a name for this kind of result?
ct.category-theory
asked 4 hours ago
Harry Gindi
8,336674160
The closest thing I can find to this is HTT 4.2.3, which considers a more general situation but can ostensibly be used in this situation. I'm looking for a more classical reference, since this seems like it should be an older result.
– Harry Gindi
4 hours ago
1
Probably it should be noted, that $textcolim_(Fdownarrow d)(Gcircpi_d)$ is a pointwise left Kan extension of $Gcirc F$ along $F$.
– Oskar
3 hours ago
@Oskar Doesn't that prove the statement then, since $operatornamecolim(operatornameLan_F Gcirc F)=operatornameLan_t_Dleft(operatornameLan_F Gcirc F right)=operatornameLan_t_D circ F Gcirc F =operatornameLan_t_CGcirc F=operatornamecolim Gcirc F$ since the terminal functor $t_C=t_Dcirc F$?
– Harry Gindi
3 hours ago
1
@HarryGingi Yes, seems that it proves this result.
– Oskar
2 hours ago
1
I added an answer, it is a summary of the comments.
– Oskar
1 hour ago
 |Â
show 1 more comment
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I could swear I remember a result of the following form:
Suppose we have a pair of functors $$CxrightarrowFDxrightarrowG X,$$ with $X$ cocomplete.
then we obtain a functor $$Dto X$$ sending $$dmapsto operatornamecolim_(Fdownarrow d)(Gcirc pi_d)$$ where $pi_d:(Fdownarrow d)to D$ is the projection functor sending $(F(c)to d) mapsto F(c)$.
Then there is a canonical isomorphism $$operatornamecolim_Dleft (operatornamecolim_(Fdownarrow d)(Gcirc pi_d)right) cong operatornamecolim_C (Gcirc F).$$
Is this true? Do you know of a source? Is there a name for this kind of result?
ct.category-theory
asked 4 hours ago
Harry Gindi
8,336674160
I could swear I remember a result of the following form:
Suppose we have a pair of functors $$CxrightarrowFDxrightarrowG X,$$ with $X$ cocomplete.
then we obtain a functor $$Dto X$$ sending $$dmapsto operatornamecolim_(Fdownarrow d)(Gcirc pi_d)$$ where $pi_d:(Fdownarrow d)to D$ is the projection functor sending $(F(c)to d) mapsto F(c)$.
Then there is a canonical isomorphism $$operatornamecolim_Dleft (operatornamecolim_(Fdownarrow d)(Gcirc pi_d)right) cong operatornamecolim_C (Gcirc F).$$
Is this true? Do you know of a source? Is there a name for this kind of result?
ct.category-theory
ct.category-theory
asked 4 hours ago
Harry Gindi
8,336674160
asked 4 hours ago
Harry Gindi
8,336674160
edited 4 hours ago
asked 4 hours ago
Harry Gindi
8,336674160
asked 4 hours ago
Harry Gindi
8,336674160
asked 4 hours ago
Harry Gindi
8,336674160
8,336674160
The closest thing I can find to this is HTT 4.2.3, which considers a more general situation but can ostensibly be used in this situation. I'm looking for a more classical reference, since this seems like it should be an older result.
– Harry Gindi
4 hours ago
1
Probably it should be noted, that $textcolim_(Fdownarrow d)(Gcircpi_d)$ is a pointwise left Kan extension of $Gcirc F$ along $F$.
– Oskar
3 hours ago
@Oskar Doesn't that prove the statement then, since $operatornamecolim(operatornameLan_F Gcirc F)=operatornameLan_t_Dleft(operatornameLan_F Gcirc F right)=operatornameLan_t_D circ F Gcirc F =operatornameLan_t_CGcirc F=operatornamecolim Gcirc F$ since the terminal functor $t_C=t_Dcirc F$?
– Harry Gindi
3 hours ago
1
@HarryGingi Yes, seems that it proves this result.
– Oskar
2 hours ago
1
I added an answer, it is a summary of the comments.
– Oskar
1 hour ago
 |Â
show 1 more comment
The closest thing I can find to this is HTT 4.2.3, which considers a more general situation but can ostensibly be used in this situation. I'm looking for a more classical reference, since this seems like it should be an older result.
– Harry Gindi
4 hours ago
1
Probably it should be noted, that $textcolim_(Fdownarrow d)(Gcircpi_d)$ is a pointwise left Kan extension of $Gcirc F$ along $F$.
– Oskar
3 hours ago
@Oskar Doesn't that prove the statement then, since $operatornamecolim(operatornameLan_F Gcirc F)=operatornameLan_t_Dleft(operatornameLan_F Gcirc F right)=operatornameLan_t_D circ F Gcirc F =operatornameLan_t_CGcirc F=operatornamecolim Gcirc F$ since the terminal functor $t_C=t_Dcirc F$?
– Harry Gindi
3 hours ago
1
@HarryGingi Yes, seems that it proves this result.
– Oskar
2 hours ago
1
I added an answer, it is a summary of the comments.
– Oskar
1 hour ago
The closest thing I can find to this is HTT 4.2.3, which considers a more general situation but can ostensibly be used in this situation. I'm looking for a more classical reference, since this seems like it should be an older result.
– Harry Gindi
4 hours ago
The closest thing I can find to this is HTT 4.2.3, which considers a more general situation but can ostensibly be used in this situation. I'm looking for a more classical reference, since this seems like it should be an older result.
– Harry Gindi
4 hours ago
1
1
Probably it should be noted, that $textcolim_(Fdownarrow d)(Gcircpi_d)$ is a pointwise left Kan extension of $Gcirc F$ along $F$.
– Oskar
3 hours ago
Probably it should be noted, that $textcolim_(Fdownarrow d)(Gcircpi_d)$ is a pointwise left Kan extension of $Gcirc F$ along $F$.
– Oskar
3 hours ago
@Oskar Doesn't that prove the statement then, since $operatornamecolim(operatornameLan_F Gcirc F)=operatornameLan_t_Dleft(operatornameLan_F Gcirc F right)=operatornameLan_t_D circ F Gcirc F =operatornameLan_t_CGcirc F=operatornamecolim Gcirc F$ since the terminal functor $t_C=t_Dcirc F$?
– Harry Gindi
3 hours ago
@Oskar Doesn't that prove the statement then, since $operatornamecolim(operatornameLan_F Gcirc F)=operatornameLan_t_Dleft(operatornameLan_F Gcirc F right)=operatornameLan_t_D circ F Gcirc F =operatornameLan_t_CGcirc F=operatornamecolim Gcirc F$ since the terminal functor $t_C=t_Dcirc F$?
– Harry Gindi
3 hours ago
1
1
@HarryGingi Yes, seems that it proves this result.
– Oskar
2 hours ago
@HarryGingi Yes, seems that it proves this result.
– Oskar
2 hours ago
1
1
I added an answer, it is a summary of the comments.
– Oskar
1 hour ago
I added an answer, it is a summary of the comments.
– Oskar
1 hour ago
 |Â
show 1 more comment
1 Answer
1
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oldest
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up vote
2
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The functor $textcolim_(Fdownarrow -)(Gcircpi_-)colon Dto X$ is a left Kan extension of $Gcirc F$ along $F$. The corresponding natural transformation $ell_F^Gcirc Fcolon Gcirc Fto textcolim_(Fdownarrow -)(Gcircpi_-)circ F$ is defined by
$$
ell_F^Gcirc F(c)=varphi^F(c)(id_F(c)),
$$
where $varphi^F(c)$ is a colimiting cocone of $Gcircpi_F(c)$. Verification of the universality is long but straightforward.
Then it remains to note that for any functors $T$ and $S$ there exists an isomorphism
$$
varinjlimtextLan_TScongvarinjlim S.
$$
Seems that your statement is just one of the great amount of interesting properties of Kan extensions, so I doubt that it has a name. Some basic properties of pointwise Kan extensions are described in the first volume of Borceux's handbook.
answered 1 hour ago
Oskar
251128
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
The functor $textcolim_(Fdownarrow -)(Gcircpi_-)colon Dto X$ is a left Kan extension of $Gcirc F$ along $F$. The corresponding natural transformation $ell_F^Gcirc Fcolon Gcirc Fto textcolim_(Fdownarrow -)(Gcircpi_-)circ F$ is defined by
$$
ell_F^Gcirc F(c)=varphi^F(c)(id_F(c)),
$$
where $varphi^F(c)$ is a colimiting cocone of $Gcircpi_F(c)$. Verification of the universality is long but straightforward.
Then it remains to note that for any functors $T$ and $S$ there exists an isomorphism
$$
varinjlimtextLan_TScongvarinjlim S.
$$
Seems that your statement is just one of the great amount of interesting properties of Kan extensions, so I doubt that it has a name. Some basic properties of pointwise Kan extensions are described in the first volume of Borceux's handbook.
answered 1 hour ago
Oskar
251128
add a comment |Â
up vote
2
down vote
The functor $textcolim_(Fdownarrow -)(Gcircpi_-)colon Dto X$ is a left Kan extension of $Gcirc F$ along $F$. The corresponding natural transformation $ell_F^Gcirc Fcolon Gcirc Fto textcolim_(Fdownarrow -)(Gcircpi_-)circ F$ is defined by
$$
ell_F^Gcirc F(c)=varphi^F(c)(id_F(c)),
$$
where $varphi^F(c)$ is a colimiting cocone of $Gcircpi_F(c)$. Verification of the universality is long but straightforward.
Then it remains to note that for any functors $T$ and $S$ there exists an isomorphism
$$
varinjlimtextLan_TScongvarinjlim S.
$$
Seems that your statement is just one of the great amount of interesting properties of Kan extensions, so I doubt that it has a name. Some basic properties of pointwise Kan extensions are described in the first volume of Borceux's handbook.
answered 1 hour ago
Oskar
251128
add a comment |Â
up vote
2
down vote
up vote
2
down vote
The functor $textcolim_(Fdownarrow -)(Gcircpi_-)colon Dto X$ is a left Kan extension of $Gcirc F$ along $F$. The corresponding natural transformation $ell_F^Gcirc Fcolon Gcirc Fto textcolim_(Fdownarrow -)(Gcircpi_-)circ F$ is defined by
$$
ell_F^Gcirc F(c)=varphi^F(c)(id_F(c)),
$$
where $varphi^F(c)$ is a colimiting cocone of $Gcircpi_F(c)$. Verification of the universality is long but straightforward.
Then it remains to note that for any functors $T$ and $S$ there exists an isomorphism
$$
varinjlimtextLan_TScongvarinjlim S.
$$
Seems that your statement is just one of the great amount of interesting properties of Kan extensions, so I doubt that it has a name. Some basic properties of pointwise Kan extensions are described in the first volume of Borceux's handbook.
answered 1 hour ago
Oskar
251128
The functor $textcolim_(Fdownarrow -)(Gcircpi_-)colon Dto X$ is a left Kan extension of $Gcirc F$ along $F$. The corresponding natural transformation $ell_F^Gcirc Fcolon Gcirc Fto textcolim_(Fdownarrow -)(Gcircpi_-)circ F$ is defined by
$$
ell_F^Gcirc F(c)=varphi^F(c)(id_F(c)),
$$
where $varphi^F(c)$ is a colimiting cocone of $Gcircpi_F(c)$. Verification of the universality is long but straightforward.
Then it remains to note that for any functors $T$ and $S$ there exists an isomorphism
$$
varinjlimtextLan_TScongvarinjlim S.
$$
Seems that your statement is just one of the great amount of interesting properties of Kan extensions, so I doubt that it has a name. Some basic properties of pointwise Kan extensions are described in the first volume of Borceux's handbook.
answered 1 hour ago
Oskar
251128
edited 1 hour ago
answered 1 hour ago
Oskar
251128
answered 1 hour ago
Oskar
251128
answered 1 hour ago
Oskar
251128
251128
add a comment |Â
add a comment |Â
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The closest thing I can find to this is HTT 4.2.3, which considers a more general situation but can ostensibly be used in this situation. I'm looking for a more classical reference, since this seems like it should be an older result.
– Harry Gindi
4 hours ago
1
Probably it should be noted, that $textcolim_(Fdownarrow d)(Gcircpi_d)$ is a pointwise left Kan extension of $Gcirc F$ along $F$.
– Oskar
3 hours ago
@Oskar Doesn't that prove the statement then, since $operatornamecolim(operatornameLan_F Gcirc F)=operatornameLan_t_Dleft(operatornameLan_F Gcirc F right)=operatornameLan_t_D circ F Gcirc F =operatornameLan_t_CGcirc F=operatornamecolim Gcirc F$ since the terminal functor $t_C=t_Dcirc F$?
– Harry Gindi
3 hours ago
1
@HarryGingi Yes, seems that it proves this result.
– Oskar
2 hours ago
1
I added an answer, it is a summary of the comments.
– Oskar
1 hour ago