Show that a functor which preserves colimits has a right adjoint
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In Moerdijk, Classifying spaces and classifying topoi, page 22, we find the following statement: a functor between topoi which preserves colimits must have a right adjoint, necessarily unique up to isomorphism (MacLane, Categories for the Working Mathematician, page 83).
Despite the reference, I actually fail to see the motivation for this. Can someone give me a hint for a proof (or an indication on how the reference is related to the problem)? It seems to me that the Theorem in CWM only covers the "uniqueness" part...
Thank you in advance.
category-theory adjoint-functors
asked Jan 14 at 21:01
W. Rether
825416
This question had a bounty worth +50
reputation from W. Rether that ended 17 hours ago. Grace period ends in 6 hours
The current answers do not contain enough detail.
A detail in the second answer, which I would need to complete the proof, is missing.
add a comment |
up vote
8
down vote
favorite
In Moerdijk, Classifying spaces and classifying topoi, page 22, we find the following statement: a functor between topoi which preserves colimits must have a right adjoint, necessarily unique up to isomorphism (MacLane, Categories for the Working Mathematician, page 83).
Despite the reference, I actually fail to see the motivation for this. Can someone give me a hint for a proof (or an indication on how the reference is related to the problem)? It seems to me that the Theorem in CWM only covers the "uniqueness" part...
Thank you in advance.
category-theory adjoint-functors
asked Jan 14 at 21:01
W. Rether
825416
This question had a bounty worth +50
reputation from W. Rether that ended 17 hours ago. Grace period ends in 6 hours
The current answers do not contain enough detail.
A detail in the second answer, which I would need to complete the proof, is missing.
add a comment |
up vote
8
down vote
favorite
up vote
8
down vote
favorite
In Moerdijk, Classifying spaces and classifying topoi, page 22, we find the following statement: a functor between topoi which preserves colimits must have a right adjoint, necessarily unique up to isomorphism (MacLane, Categories for the Working Mathematician, page 83).
Despite the reference, I actually fail to see the motivation for this. Can someone give me a hint for a proof (or an indication on how the reference is related to the problem)? It seems to me that the Theorem in CWM only covers the "uniqueness" part...
Thank you in advance.
category-theory adjoint-functors
asked Jan 14 at 21:01
W. Rether
825416
In Moerdijk, Classifying spaces and classifying topoi, page 22, we find the following statement: a functor between topoi which preserves colimits must have a right adjoint, necessarily unique up to isomorphism (MacLane, Categories for the Working Mathematician, page 83).
Despite the reference, I actually fail to see the motivation for this. Can someone give me a hint for a proof (or an indication on how the reference is related to the problem)? It seems to me that the Theorem in CWM only covers the "uniqueness" part...
Thank you in advance.
category-theory adjoint-functors
category-theory adjoint-functors
asked Jan 14 at 21:01
W. Rether
825416
asked Jan 14 at 21:01
W. Rether
825416
edited Jan 14 at 21:13
asked Jan 14 at 21:01
W. Rether
825416
asked Jan 14 at 21:01
W. Rether
825416
asked Jan 14 at 21:01
W. Rether
825416
825416
This question had a bounty worth +50
reputation from W. Rether that ended 17 hours ago. Grace period ends in 6 hours
The current answers do not contain enough detail.
A detail in the second answer, which I would need to complete the proof, is missing.
This question had a bounty worth +50
reputation from W. Rether that ended 17 hours ago. Grace period ends in 6 hours
The current answers do not contain enough detail.
A detail in the second answer, which I would need to complete the proof, is missing.
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
up vote
2
down vote
accepted
I don't know a reference for elementary topoi, but one for presheaf topoi can be found around pages 41-43 of MacLane-Moerdijk Sheaves in Geometry and Logic: they prove (Corollary 4) that a functor $A:mathbb{C}rightarrowmathcal{E}$ where $mathcal{E}$ is cocomplete and $mathbb{C}$ is small extends along the Yoneda embedding $y$ to a unique (up to iso) $L_A:widehat{mathbb{C}}rightarrowmathcal{E}$ which preserves colimits, and such a functor has a right adjoint by construction (Theorem 2).
If now you consider some colimit preserving functor $F:widehat{mathbb{C}}rightarrowwidehat{mathbb{D}}$, the composition $A=Fy$ must extend in such a way: so it extends to $L_A=F$, but then $F$ must have a right adjoint.
answered Jan 14 at 21:33
TheMadcapLaughs
381211
4
The theorem is false for elementary topoi, basically since they can be small: consider the inclusion of finite sets into sets.
– Kevin Carlson
Jan 15 at 18:44
Just a question: I understand that by $hat{mathbb C}$ you mean presheaves. What if I have sheaves? It is not always true that $mathbb C$ embeds in $Sh(mathbb C,J)$, i.e. not every topology is subcanonical.
– W. Rether
Nov 9 at 9:24
Update: using the adjoint pairs (inclusion, sheafification) between the topos of sheaves and the category of presheaves solves the problem. Thanks!
– W. Rether
Nov 18 at 12:14
add a comment |
up vote
9
down vote
This is false in general without further hypotheses; see adjoint functor theorem. The reference to CWM, as you say, is only a reference for the uniqueness. The various adjoint functor theorems do imply this statement for a functor between Grothendieck topoi.
answered Jan 14 at 21:05
Qiaochu Yuan
274k32578914
Thank you. Indeed, the context I'm interested in is that of topoi. Edited!
– W. Rether
Jan 14 at 21:13
1
In particular, the nLab mentions (at the bottom of the Statements section) a fairly general result about functors between locally presentable categories which states any functor between locally presentable categories is a left adjoint if and only if it preserves colimits. This includes sheaf toposes but also many other handy things like categories of algebras.
– Derek Elkins
Jan 14 at 23:11
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
I don't know a reference for elementary topoi, but one for presheaf topoi can be found around pages 41-43 of MacLane-Moerdijk Sheaves in Geometry and Logic: they prove (Corollary 4) that a functor $A:mathbb{C}rightarrowmathcal{E}$ where $mathcal{E}$ is cocomplete and $mathbb{C}$ is small extends along the Yoneda embedding $y$ to a unique (up to iso) $L_A:widehat{mathbb{C}}rightarrowmathcal{E}$ which preserves colimits, and such a functor has a right adjoint by construction (Theorem 2).
If now you consider some colimit preserving functor $F:widehat{mathbb{C}}rightarrowwidehat{mathbb{D}}$, the composition $A=Fy$ must extend in such a way: so it extends to $L_A=F$, but then $F$ must have a right adjoint.
answered Jan 14 at 21:33
TheMadcapLaughs
381211
4
The theorem is false for elementary topoi, basically since they can be small: consider the inclusion of finite sets into sets.
– Kevin Carlson
Jan 15 at 18:44
Just a question: I understand that by $hat{mathbb C}$ you mean presheaves. What if I have sheaves? It is not always true that $mathbb C$ embeds in $Sh(mathbb C,J)$, i.e. not every topology is subcanonical.
– W. Rether
Nov 9 at 9:24
Update: using the adjoint pairs (inclusion, sheafification) between the topos of sheaves and the category of presheaves solves the problem. Thanks!
– W. Rether
Nov 18 at 12:14
add a comment |
up vote
2
down vote
accepted
I don't know a reference for elementary topoi, but one for presheaf topoi can be found around pages 41-43 of MacLane-Moerdijk Sheaves in Geometry and Logic: they prove (Corollary 4) that a functor $A:mathbb{C}rightarrowmathcal{E}$ where $mathcal{E}$ is cocomplete and $mathbb{C}$ is small extends along the Yoneda embedding $y$ to a unique (up to iso) $L_A:widehat{mathbb{C}}rightarrowmathcal{E}$ which preserves colimits, and such a functor has a right adjoint by construction (Theorem 2).
If now you consider some colimit preserving functor $F:widehat{mathbb{C}}rightarrowwidehat{mathbb{D}}$, the composition $A=Fy$ must extend in such a way: so it extends to $L_A=F$, but then $F$ must have a right adjoint.
answered Jan 14 at 21:33
TheMadcapLaughs
381211
4
The theorem is false for elementary topoi, basically since they can be small: consider the inclusion of finite sets into sets.
– Kevin Carlson
Jan 15 at 18:44
Just a question: I understand that by $hat{mathbb C}$ you mean presheaves. What if I have sheaves? It is not always true that $mathbb C$ embeds in $Sh(mathbb C,J)$, i.e. not every topology is subcanonical.
– W. Rether
Nov 9 at 9:24
Update: using the adjoint pairs (inclusion, sheafification) between the topos of sheaves and the category of presheaves solves the problem. Thanks!
– W. Rether
Nov 18 at 12:14
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
I don't know a reference for elementary topoi, but one for presheaf topoi can be found around pages 41-43 of MacLane-Moerdijk Sheaves in Geometry and Logic: they prove (Corollary 4) that a functor $A:mathbb{C}rightarrowmathcal{E}$ where $mathcal{E}$ is cocomplete and $mathbb{C}$ is small extends along the Yoneda embedding $y$ to a unique (up to iso) $L_A:widehat{mathbb{C}}rightarrowmathcal{E}$ which preserves colimits, and such a functor has a right adjoint by construction (Theorem 2).
If now you consider some colimit preserving functor $F:widehat{mathbb{C}}rightarrowwidehat{mathbb{D}}$, the composition $A=Fy$ must extend in such a way: so it extends to $L_A=F$, but then $F$ must have a right adjoint.
answered Jan 14 at 21:33
TheMadcapLaughs
381211
I don't know a reference for elementary topoi, but one for presheaf topoi can be found around pages 41-43 of MacLane-Moerdijk Sheaves in Geometry and Logic: they prove (Corollary 4) that a functor $A:mathbb{C}rightarrowmathcal{E}$ where $mathcal{E}$ is cocomplete and $mathbb{C}$ is small extends along the Yoneda embedding $y$ to a unique (up to iso) $L_A:widehat{mathbb{C}}rightarrowmathcal{E}$ which preserves colimits, and such a functor has a right adjoint by construction (Theorem 2).
If now you consider some colimit preserving functor $F:widehat{mathbb{C}}rightarrowwidehat{mathbb{D}}$, the composition $A=Fy$ must extend in such a way: so it extends to $L_A=F$, but then $F$ must have a right adjoint.
answered Jan 14 at 21:33
TheMadcapLaughs
381211
answered Jan 14 at 21:33
TheMadcapLaughs
381211
answered Jan 14 at 21:33
TheMadcapLaughs
381211
answered Jan 14 at 21:33
TheMadcapLaughs
381211
381211
4
The theorem is false for elementary topoi, basically since they can be small: consider the inclusion of finite sets into sets.
– Kevin Carlson
Jan 15 at 18:44
Just a question: I understand that by $hat{mathbb C}$ you mean presheaves. What if I have sheaves? It is not always true that $mathbb C$ embeds in $Sh(mathbb C,J)$, i.e. not every topology is subcanonical.
– W. Rether
Nov 9 at 9:24
Update: using the adjoint pairs (inclusion, sheafification) between the topos of sheaves and the category of presheaves solves the problem. Thanks!
– W. Rether
Nov 18 at 12:14
add a comment |
4
The theorem is false for elementary topoi, basically since they can be small: consider the inclusion of finite sets into sets.
– Kevin Carlson
Jan 15 at 18:44
Just a question: I understand that by $hat{mathbb C}$ you mean presheaves. What if I have sheaves? It is not always true that $mathbb C$ embeds in $Sh(mathbb C,J)$, i.e. not every topology is subcanonical.
– W. Rether
Nov 9 at 9:24
Update: using the adjoint pairs (inclusion, sheafification) between the topos of sheaves and the category of presheaves solves the problem. Thanks!
– W. Rether
Nov 18 at 12:14
4
4
The theorem is false for elementary topoi, basically since they can be small: consider the inclusion of finite sets into sets.
– Kevin Carlson
Jan 15 at 18:44
The theorem is false for elementary topoi, basically since they can be small: consider the inclusion of finite sets into sets.
– Kevin Carlson
Jan 15 at 18:44
Just a question: I understand that by $hat{mathbb C}$ you mean presheaves. What if I have sheaves? It is not always true that $mathbb C$ embeds in $Sh(mathbb C,J)$, i.e. not every topology is subcanonical.
– W. Rether
Nov 9 at 9:24
Just a question: I understand that by $hat{mathbb C}$ you mean presheaves. What if I have sheaves? It is not always true that $mathbb C$ embeds in $Sh(mathbb C,J)$, i.e. not every topology is subcanonical.
– W. Rether
Nov 9 at 9:24
Update: using the adjoint pairs (inclusion, sheafification) between the topos of sheaves and the category of presheaves solves the problem. Thanks!
– W. Rether
Nov 18 at 12:14
Update: using the adjoint pairs (inclusion, sheafification) between the topos of sheaves and the category of presheaves solves the problem. Thanks!
– W. Rether
Nov 18 at 12:14
add a comment |
up vote
9
down vote
This is false in general without further hypotheses; see adjoint functor theorem. The reference to CWM, as you say, is only a reference for the uniqueness. The various adjoint functor theorems do imply this statement for a functor between Grothendieck topoi.
answered Jan 14 at 21:05
Qiaochu Yuan
274k32578914
Thank you. Indeed, the context I'm interested in is that of topoi. Edited!
– W. Rether
Jan 14 at 21:13
1
In particular, the nLab mentions (at the bottom of the Statements section) a fairly general result about functors between locally presentable categories which states any functor between locally presentable categories is a left adjoint if and only if it preserves colimits. This includes sheaf toposes but also many other handy things like categories of algebras.
– Derek Elkins
Jan 14 at 23:11
add a comment |
up vote
9
down vote
This is false in general without further hypotheses; see adjoint functor theorem. The reference to CWM, as you say, is only a reference for the uniqueness. The various adjoint functor theorems do imply this statement for a functor between Grothendieck topoi.
answered Jan 14 at 21:05
Qiaochu Yuan
274k32578914
Thank you. Indeed, the context I'm interested in is that of topoi. Edited!
– W. Rether
Jan 14 at 21:13
1
In particular, the nLab mentions (at the bottom of the Statements section) a fairly general result about functors between locally presentable categories which states any functor between locally presentable categories is a left adjoint if and only if it preserves colimits. This includes sheaf toposes but also many other handy things like categories of algebras.
– Derek Elkins
Jan 14 at 23:11
add a comment |
up vote
9
down vote
up vote
9
down vote
This is false in general without further hypotheses; see adjoint functor theorem. The reference to CWM, as you say, is only a reference for the uniqueness. The various adjoint functor theorems do imply this statement for a functor between Grothendieck topoi.
answered Jan 14 at 21:05
Qiaochu Yuan
274k32578914
This is false in general without further hypotheses; see adjoint functor theorem. The reference to CWM, as you say, is only a reference for the uniqueness. The various adjoint functor theorems do imply this statement for a functor between Grothendieck topoi.
answered Jan 14 at 21:05
Qiaochu Yuan
274k32578914
edited Jan 14 at 21:14
answered Jan 14 at 21:05
Qiaochu Yuan
274k32578914
answered Jan 14 at 21:05
Qiaochu Yuan
274k32578914
answered Jan 14 at 21:05
Qiaochu Yuan
274k32578914
274k32578914
Thank you. Indeed, the context I'm interested in is that of topoi. Edited!
– W. Rether
Jan 14 at 21:13
1
In particular, the nLab mentions (at the bottom of the Statements section) a fairly general result about functors between locally presentable categories which states any functor between locally presentable categories is a left adjoint if and only if it preserves colimits. This includes sheaf toposes but also many other handy things like categories of algebras.
– Derek Elkins
Jan 14 at 23:11
add a comment |
Thank you. Indeed, the context I'm interested in is that of topoi. Edited!
– W. Rether
Jan 14 at 21:13
1
In particular, the nLab mentions (at the bottom of the Statements section) a fairly general result about functors between locally presentable categories which states any functor between locally presentable categories is a left adjoint if and only if it preserves colimits. This includes sheaf toposes but also many other handy things like categories of algebras.
– Derek Elkins
Jan 14 at 23:11
Thank you. Indeed, the context I'm interested in is that of topoi. Edited!
– W. Rether
Jan 14 at 21:13
Thank you. Indeed, the context I'm interested in is that of topoi. Edited!
– W. Rether
Jan 14 at 21:13
1
1
In particular, the nLab mentions (at the bottom of the Statements section) a fairly general result about functors between locally presentable categories which states any functor between locally presentable categories is a left adjoint if and only if it preserves colimits. This includes sheaf toposes but also many other handy things like categories of algebras.
– Derek Elkins
Jan 14 at 23:11
In particular, the nLab mentions (at the bottom of the Statements section) a fairly general result about functors between locally presentable categories which states any functor between locally presentable categories is a left adjoint if and only if it preserves colimits. This includes sheaf toposes but also many other handy things like categories of algebras.
– Derek Elkins
Jan 14 at 23:11
add a comment |
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