Factorization of colimits through slices?
Clash Royale CLAN TAG#URR8PPP
up vote
5
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
 |Â
show 1 more comment
up vote
5
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
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.
â Harry Gindi
Sep 17 at 1:20
1
Probably it should be noted, that $textcolim_(Fdownarrow d)(Gcircpi_d)$ is a pointwise left Kan extension of $Gcirc F$ along $F$.
â Oskar
Sep 17 at 1:31
@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 $t_C=t_Dcirc F$?
â Harry Gindi
Sep 17 at 2:00
1
@HarryGingi Yes, seems that it proves this result.
â Oskar
Sep 17 at 2:23
2
I added an answer, it is a summary of the comments.
â Oskar
Sep 17 at 4:07
 |Â
show 1 more comment
up vote
5
down vote
favorite
up vote
5
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
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
edited Sep 17 at 1:03
asked Sep 17 at 0:56
Harry Gindi
8,368674160
8,368674160
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.
â Harry Gindi
Sep 17 at 1:20
1
Probably it should be noted, that $textcolim_(Fdownarrow d)(Gcircpi_d)$ is a pointwise left Kan extension of $Gcirc F$ along $F$.
â Oskar
Sep 17 at 1:31
@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 $t_C=t_Dcirc F$?
â Harry Gindi
Sep 17 at 2:00
1
@HarryGingi Yes, seems that it proves this result.
â Oskar
Sep 17 at 2:23
2
I added an answer, it is a summary of the comments.
â Oskar
Sep 17 at 4:07
 |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.
â Harry Gindi
Sep 17 at 1:20
1
Probably it should be noted, that $textcolim_(Fdownarrow d)(Gcircpi_d)$ is a pointwise left Kan extension of $Gcirc F$ along $F$.
â Oskar
Sep 17 at 1:31
@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 $t_C=t_Dcirc F$?
â Harry Gindi
Sep 17 at 2:00
1
@HarryGingi Yes, seems that it proves this result.
â Oskar
Sep 17 at 2:23
2
I added an answer, it is a summary of the comments.
â Oskar
Sep 17 at 4:07
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.
â Harry Gindi
Sep 17 at 1:20
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.
â Harry Gindi
Sep 17 at 1:20
1
1
Probably it should be noted, that $textcolim_(Fdownarrow d)(Gcircpi_d)$ is a pointwise left Kan extension of $Gcirc F$ along $F$.
â Oskar
Sep 17 at 1:31
Probably it should be noted, that $textcolim_(Fdownarrow d)(Gcircpi_d)$ is a pointwise left Kan extension of $Gcirc F$ along $F$.
â Oskar
Sep 17 at 1:31
@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 $t_C=t_Dcirc F$?
â Harry Gindi
Sep 17 at 2:00
@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 $t_C=t_Dcirc F$?
â Harry Gindi
Sep 17 at 2:00
1
1
@HarryGingi Yes, seems that it proves this result.
â Oskar
Sep 17 at 2:23
@HarryGingi Yes, seems that it proves this result.
â Oskar
Sep 17 at 2:23
2
2
I added an answer, it is a summary of the comments.
â Oskar
Sep 17 at 4:07
I added an answer, it is a summary of the comments.
â Oskar
Sep 17 at 4:07
 |Â
show 1 more comment
1 Answer
1
active
oldest
votes
up vote
6
down vote
accepted
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.
1
This is in fact a result about compatibility of different weights in weighted colimits, valid more generally in enriched/formal category theory. See for example Riehl's Categorical Homotopy Theory, Lemma 8.1.4.
â Fosco Loregian
Sep 17 at 20:40
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
down vote
accepted
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.
1
This is in fact a result about compatibility of different weights in weighted colimits, valid more generally in enriched/formal category theory. See for example Riehl's Categorical Homotopy Theory, Lemma 8.1.4.
â Fosco Loregian
Sep 17 at 20:40
add a comment |Â
up vote
6
down vote
accepted
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.
1
This is in fact a result about compatibility of different weights in weighted colimits, valid more generally in enriched/formal category theory. See for example Riehl's Categorical Homotopy Theory, Lemma 8.1.4.
â Fosco Loregian
Sep 17 at 20:40
add a comment |Â
up vote
6
down vote
accepted
up vote
6
down vote
accepted
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.
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.
edited Sep 17 at 4:20
answered Sep 17 at 4:05
Oskar
306128
306128
1
This is in fact a result about compatibility of different weights in weighted colimits, valid more generally in enriched/formal category theory. See for example Riehl's Categorical Homotopy Theory, Lemma 8.1.4.
â Fosco Loregian
Sep 17 at 20:40
add a comment |Â
1
This is in fact a result about compatibility of different weights in weighted colimits, valid more generally in enriched/formal category theory. See for example Riehl's Categorical Homotopy Theory, Lemma 8.1.4.
â Fosco Loregian
Sep 17 at 20:40
1
1
This is in fact a result about compatibility of different weights in weighted colimits, valid more generally in enriched/formal category theory. See for example Riehl's Categorical Homotopy Theory, Lemma 8.1.4.
â Fosco Loregian
Sep 17 at 20:40
This is in fact a result about compatibility of different weights in weighted colimits, valid more generally in enriched/formal category theory. See for example Riehl's Categorical Homotopy Theory, Lemma 8.1.4.
â Fosco Loregian
Sep 17 at 20:40
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f310730%2ffactorization-of-colimits-through-slices%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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.
â Harry Gindi
Sep 17 at 1:20
1
Probably it should be noted, that $textcolim_(Fdownarrow d)(Gcircpi_d)$ is a pointwise left Kan extension of $Gcirc F$ along $F$.
â Oskar
Sep 17 at 1:31
@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 $t_C=t_Dcirc F$?
â Harry Gindi
Sep 17 at 2:00
1
@HarryGingi Yes, seems that it proves this result.
â Oskar
Sep 17 at 2:23
2
I added an answer, it is a summary of the comments.
â Oskar
Sep 17 at 4:07