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edited Nov 17 2009 at 9:08 |
Yes, and it is easy I think frequently the easiest to check . The sufficient conditions are the followingresult makes , which also make precise why "this looks like it might work" is so often successful:
Theorem: A functor $G: C\to D$ is a right adjoint functor (i.e. has a left adjoint) if and only if for each object $Y$ in $D$, there exists an initial morphism $\phi_Y:Y\to G(I_Y)$ from $Y$ to $G$. Moreover, once you find such an initial morphism from each $Y$ to $G$, the association $Y\mapsto I_Y$ extends in a unique way to act on morphisms defining a functor $F: D\to C$, which moreover is left adjoint to the original functor $G$.
This is well-known and easy to prove (well, depending on who you ask), but is non-trivial and involves many steps, which are explained relatively well here. (Essentially one is recovering the entire adjoint situation from just one functor and a unit transformation.)
Once you know it, you can really take confidence in "follow your nose"-style adjoint construction. It doesn't involve having an "initial guess" for the left adjoint (as a functor), but actually constructs it for you in a way that is uniquely determined by the limited data of the initial morphisms --- really unique, not just up to natural isomorphism.
As an example of how this can be useful, think of the inclusion functor $U$ from $AbGrp$ to $Grp$. It's easy to see that any group $H$ has an abelianization $Ab(H) = H/[H,H]$ in $AbGrp$ with a map
$H\to Ab(H)$ satisfying an initial (universal) property. But then by the above theorem, we can automatically extend this association in a unique way to act on morphisms as well, defining an abelianization functor $Ab$ which is left adjoint to the inclusion $U$.
This same trick expedites the construction of adjoints in pretty much any situation you can think of.
Edit: Sometimes this theorem is used as an alternative definition for adjoint functors in terms of universal morphisms. However you look at it, the real utility is knowing that this "weak", and in fact asymetric, condition actually implies the "stronger", symmetric definitions of adjoints via hom-sets or units/counits. I think it's really worthwhile to sift through the three different characterizations of adjoints given on Wikipedia.
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edited Nov 17 2009 at 8:57 |
Yes, and it is easy to check! In fact, the . The following theorem result makes precise why "this looks like it might work" is so often successful: This result is well-known , and easy to prove (well, depending on who you ask), but is non-trivial and involves many steps, which are explained relatively well here. (Essentially one is recovering the entire adjoint situation from just one functor and a unit transformation.) Once you know it, you can really take confidence in "follow your nose"-style adjoint construction. It doesn't involve having an "initial guess" for the left adjoint (as a functor), but actually constructs it for you in a way that is uniquely determined by the limited data of the initial morphisms --- really unique, not just up to natural isomorphism. This same trick expedites the construction of adjoints in pretty much any situation you can think of. I think it's really worthwhile to sift through the three different characterizations of adjoints given on Wikipedia. Notice that: (1) This theorem does not involve having an "initial guess" for the left adjoint (as a functor), but actually constructs it for you in a way that is uniquely determined by the limited data of the initial morphisms (I think it's really unique, not just up worthwhile to natural isomorphism). (2) Its conditions are asymmetric, but the conclusion and sift through the definition three different characterizations of adjoints (using hom-sets) is symmetric, which means there really is something to prove here. (3) It follows further from this theorem that the functor $G$ actually defines provides you with a family of terminal morphisms to the functor $F$, which is now many logical steps removed from the assumption that initial morphisms existed from $G$. (4) The initial morphisms in the theorem end up forming the unit of the adjunction, but we do not start off assuming these morphisms are natural tranformations. Instead, the functor $F$ is constructed in a way to make them natural, and the counit (ie terminal morphisms) end up existing for free, and satisfy the required equationsgiven on Wikipedia.
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edited Nov 17 2009 at 8:48 |
Yes, and it is easy to check! In fact, the following theorem makes precise why "this looks like it might work" is so often successful:
Theorem: A functor $G: C\to D$ is a right adjoint functor (i.e. has a left adjoint) if and only if for each object $Y$ in $D$, there exists an initial morphism $\phi_Y:Y\to G(I_Y)$ from $Y$ to $G$. Moreover, once you find such an initial morphism from each $Y$ to $G$, the association $Y\mapsto I_Y$ extends in a unique way to act on morphisms defining a functor $F: D\to C$, which moreover is left adjoint to the original functor $G$.
This result is well-known, easy to prove, but is non-trivial and involves many steps, which are explained relatively well here. (Essentially one is recovering the entire adjoint situation from just one functor and a unit transformation.) Once you know it, you can really take confidence in "follow your nose"-style adjoint construction.
As an example of how this can be useful, think of the inclusion functor $U$ from $AbGrp$ to $Grp$. It's easy to see that any group $H$ has an abelianization $Ab(H) = H/[H,H]$ in $AbGrp$ with a map
$H\to Ab(H)$ satisfying an initial (universal) property. But then by the above theorem, we can automatically extend this association in a unique way to act on morphisms as well, defining an abelianization functor $Ab$ which is left adjoint to the inclusion $U$.
This same trick expedites the construction of adjoints in pretty much any situation you can think of. I think it's really worthwhile to sift through the three different characterizations of adjoints given on Wikipedia.
Edit: Sometimes this theorem is used as an alternative definition for adjoint functors in terms of universal morphisms. However you look at it, the real utility is knowing that this "weak", and in fact asymetric, condition actually implies the "stronger", symmetric definitions of adjoints via hom-sets or units/counits.
Notice that:
(1) This theorem does not involve having an "initial guess" for the left adjoint (as a functor), but actually constructs it for you in a way that is uniquely determined by the limited data of the initial morphisms (really unique, not just up to natural isomorphism).
(2) Its conditions are asymmetric, but the conclusion and the definition of adjoints (using hom-sets) is symmetric, which means there really is something to prove here.
(3) It follows further from this theorem that the functor $G$ actually defines provides you with a family of terminal morphisms to the functor $F$, which is now many logical steps removed from the assumption that initial morphisms existed from $G$.
(4) The initial morphisms in the theorem end up forming the unit of the adjunction, but we do not start off assuming these morphisms are natural tranformations. Instead, the functor $F$ is constructed in a way to make them natural, and the counit (ie terminal morphisms) end up existing for free, and satisfy the required equations.
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edited Nov 17 2009 at 8:43 |
Yes, and it is easy to check! In fact, the following theorem makes precise why "this looks like it might work" is so often successful:
Theorem: A functor $G: C\to D$ is a right adjoint functor (i.e. has a left adjoint) if and only if for each object $Y$ in $D$, there exists an initial morphism $\phi_Y:Y\to G(I_Y)$ from $Y$ to $G$. Moreover, once you find such an initial morphism from each $Y$ to $G$, the association $Y\mapsto I_Y$ extends in a unique way to act on morphisms defining a functor $F: D\to C$, which moreover is left adjoint to the original functor $G$.
This result is well-known, not that hard easy to prove, but is non-trivial and involves many steps, which are explained relatively well here. A discussion of why this result is not trivial (Essentially one is appended below. recovering the entire adjoint situation from just one functor and a unit transformation.) Once you know it, you can really take confidence in "follow your nose"-style adjoint construction.
As an example of how this can be useful, think of the inclusion functor $U$ from $AbGrp$ to $Grp$. It's easy to see that any group $H$ has an abelianization $Ab(H) = H/[H,H]$ in $AbGrp$ with a map
$H\to Ab(H)$ satisfying an initial (universal) property. But then by the above theorem, we can automatically extend this association in a unique way to act on morphisms as well, defining an abelianization functor $Ab$ which is left adjoint to the inclusion $U$.
This same trick expedites the construction of adjoints in pretty much any situation you can think of. I think it's really worthwhile to sift through the three different characterizations of adjoints given on Wikipedia.
Edit: Sometimes this theorem is used as an alternative definition for adjoint functors in terms of universal morphisms. However you look at it, the real utility is knowing that this "weak" , weak", and in fact asymetric, condition actually implies the "stronger", symmetric definitions of adjoints via hom-sets or units/counits.
Notice that:
(1) This theorem does not involve having an "initial guess" for the left adjoint (as a functor), but actually constructs it for you in a way that is uniquely determined by the limited data of the initial morphisms (really unique, not just up to natural isomorphism).
(2) Its conditions are asymmetric, but the conclusion and the definition of adjoints (using hom-sets) is symmetric, which means there really is something to prove here.
(3)It follows further from this theorem that the functor $G$ actually defines provides you with a family of terminal morphisms to the functor $F$, which is now many logical steps removed from the assumption that initial morphisms existed from $G$.
(4) The initial morphisms in the theorem end up forming the unit of the adjunction, but we do not start off assuming these morphisms are natural tranformations. Instead, the functor $F$ is constructed in a way to make them natural, and the counit (ie terminal morphisms) end up existing for free, and satisfy the required equations.
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edited Nov 17 2009 at 8:34 |
It This result is well-known, not that hard to prove, but is non-trivial and involves many steps, which are explained relatively well here. A discussion of why this result is not trivial is appended below. Once you know thisit, you can really take confidence in "follow your nose"-style adjoint construction. Notice that: (1) This theorem does not involve having an "initial guess" for the left adjoint, but actually constructs it for you in a way that is uniquely determined by the limited data of the initial morphisms (really unique, not just up to natural isomorphism). (2) Its conditions are asymmetric, but conclusion and the definition of adjoints (using hom-sets) is symmetric, which means there really is something to prove here. For example Notice that: (1) This theorem does not involve having an "initial guess" for the left adjoint, but actually constructs it for you in a way that is uniquely determined by the limited data of the initial morphisms (really unique, not just up to natural isomorphism). (2) Its conditions are asymmetric, but the conclusion and the definition of adjoints (using hom-sets) is symmetric, which means there really is something to prove here. (3)It follows further from this theorem that the functor $G$ actually defines provides you with a family of terminal morphisms to the functor $F$, which is now many logical steps removed from the assumption that initial morphisms existed from $G$. (4) The initial morphisms in the theorem end up forming the unit of the adjunction, but we do not start off assuming these morphisms are natural tranformations. Instead, the functor $F$ is constructed in a way to make them natural, and the counit (ie terminal morphisms) end up existing for free, and satisfy the required equations.
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edited Nov 17 2009 at 8:16 |
Yes, and it is easy to check! In fact, the following theorem makes precise why "this looks like it might work" is so often successful:
Theorem: A functor $G: C\to D$ is a right adjoint functor (i.e. has a left adjoint) if and only if for each object $Y$ in $D$, there exists an initial morphism $\phi_Y:Y\to G(I_Y)$ from $Y$ to $G$. Moreover, once you find such an initial morphism from each $Y$ to $G$, the association $Y\mapsto I_Y$ extends in a unique way to act on morphisms defining a functor $F: D\to C$, which moreover is left adjoint to the original functor $G$.
Notice
It is not that hard to prove, but non-trivial and involves many steps, which are explained relatively well here.
Once you know this, you can really take confidence in "follow your nose"-style adjoint construction. Notice that:
(1) This theorem does not involve having an "initial guess" for the left adjoint, but actually constructs it for you in a way that is uniquely determined by the limited data of the initial morphisms (really unique, not just up to natural isomorphism). It is not hard to prove
(2) Its conditions are asymmetric, but not trivial, conclusion and the definition of adjoints (using hom-sets) is explained relatively well here.
Once you know thissymmetric, you can which means there really take confidence in "follow your nose"-style adjoint constructionis something to prove here.It doesn't even involve having an initial guess for the left adjoint (as a functor), since the theorem constructs it for you from the l
As an example of how this can be useful, think of the inclusion functor $U$ from $AbGrp$ to $Grp$. It's easy to see that any group $H$ has an abelianization $Ab(H) = H/[H,H]$ in $AbGrp$ with a map $H\to Ab(H)$ satisfying an initial (universal) property. But then by the above theorem, we can automatically extend this association in a unique way to act on morphisms as well, defining an abelianization functor $Ab$ which is left adjoint to the inclusion $U$.
This same trick expedites the construction of adjoints in pretty much any situation you can think of. I think it's really worthwhile to sift through the three different characterizations of adjoints given on Wikipedia.
Edit: Sometimes this theorem is used as an alternative definition for adjoint functors in terms of universal morphisms. However you look at it, the real utility is knowing that this "weak" , and in fact asymetric, condition actually implies the "stronger", symmetric definitions of adjoints via hom-sets or units/counits.
For example, it follows further from this theorem that the functor $G$ actually defines provides you with a family of terminal morphisms to the functor $F$, which is now many logical steps removed from the assumption that initial morphisms existed from $G$.I think the non-triviality of the various definitions of adjoint functors is a commonly overlooked phenomenon.
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edited Nov 17 2009 at 8:03 |
Yes, and it is easy to check! In fact, the following theorem makes precise why "this looks like it might work" is so often successful:
Theorem: A functor $G: C\to D$ is a right adjoint functor (i.e. has a left adjoint) if and only if for each object $Y$ in $D$, there exists an initial morphism $\phi_Y:Y\to G(I_Y)$ from $Y$ to $G$. Moreover, once you find such an initial morphism from each $Y$ to $G$, the association $Y\mapsto I_Y$ extends in a unique way to act on morphisms defining a functor $F: D\to C$, which moreover is left adjoint to the original functor $G$.
Notice that this theorem does not involve having an "initial guess" for the left adjoint, but actually constructs it for you in a way that is uniquely determined by the limited data of the initial morphisms (really unique, not just up to natural isomorphism). It is not hard to prove, but not trivial, and is explained relatively well here.
Once you know this, you can really take confidence in "follow your nose"-style adjoint construction. It doesn't even involve having an initial guess for the left adjoint (as a functor), since the theorem constructs it for you from the l
As an example of how this can be useful, think of the inclusion functor $U$ from $AbGrp$ to $Grp$. It's easy to see that any group $H$ has an abelianization $Ab(H) = H/[H,H]$ in $AbGrp$ with a map $H\to Ab(H)$ satisfying an initial (universal) property. But then by the above theorem, we can automatically extend this association in a unique way to act on morphisms as well, defining an abelianization functor $Ab$ which is left adjoint to the inclusion $U$.
This same trick expedites the construction of adjoints in pretty much any situation you can think of. I think it's really worthwhile to sift through the three different characterizations of adjoints given on Wikipedia.
Edit: Sometimes this theorem is used as an alternative definition for adjoint functors in terms of universal morphisms. However you look at it, the real utility is knowing that this "weak" , and in fact asymetric, condition actually implies the "stronger" stronger", symmetric definitions of adjoints via hom-sets or units/counits.
For example, it follows further from this theorem that the functor $G$ actually defines provides you with a family of terminal morphisms to the functor $F$, which is now many logical steps removed from the assumption that initial morphisms existed from $G$. I think the non-triviality of the various definitions of adjoint functors is a commonly overlooked phenomenon.
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edited Nov 17 2009 at 7:50 |
Yes, and it is easy to check! In fact, the following theorem makes precise why "this looks like it might work" is so often successful:
Theorem: A functor $G: C\to D$ is a right adjoint functor (i.e. has a left adjoint) if and only if for each object $Y$ in $D$, there exists an initial morphism $\phi_Y:Y\to G(I_Y)$ from $Y$ to $G$. Moreover, once you find such an initial morphism from each $Y$ to $G$, the association $Y\mapsto I_Y$ extends in a unique way to act on morphisms defining a functor $F: D\to C$, which moreover is left adjoint to the original functor $G$.(Once you have the initial morphisms, this extension is really unique, i.e. not just unique up to natural isomorphism.)
Note
Notice that this theorem does not involve having an "initial guess" for the left adjoint, but actually constructs it for you from in a way that is uniquely determined by the limited data of the initial morphisms (really unique, not just up to natural isomorphism). It is not hard to prove, but not trivial, and is explained relatively well here.
Once you know this, you can really take confidence in "follow your nose"-style adjoint construction. It doesn't even involve having an initial guess for the left adjoint (as a functor), since the theorem constructs it for you from the l
As an example of how this can be useful, think of the inclusion functor $U$ from $AbGrp$ to $Grp$. It's easy to see that any group $H$ has an abelianization $Ab(H) = H/[H,H]$ in $AbGrp$ with a map $H\to Ab(H)$ satisfying an initial (universal) property. But then by the above theorem, we can automatically extend this association in a unique way to act on morphisms as well, defining an abelianization functor $Ab$ which is left adjoint to the inclusion $U$.
This same trick expedites the construction of adjoints in pretty much any situation you can think of. I think it's really worthwhile to sift through the three different characterizations of adjoints given on Wikipedia.
Edit: Sometimes this theorem is used as an alternative definition for adjoint functors in terms of universal morphisms. However you look at it, the real utility is knowing that this "weak" condition actually implies the "stronger" definitions of adjoints via hom-sets or units/counits.
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edited Nov 17 2009 at 7:37 |
Yes, and it is easy to check! In fact, the following theorem makes precise why "this looks like it might work" is so often successful:
Theorem: A functor $G: C\to D$ is a right adjoint functor (i.e. has a left adjoint) if and only if for each object $Y$ in $D$, there exists an initial morphism $\phi_Y:Y\to G(T_Y)$ G(I_Y)$ from $Y$ to $G$. Moreover, once you find such an initial morphism from each $Y$ to $G$, the association $Y\mapsto T_Y$ I_Y$ extends in a unique way (really unique, i.e. not just up to natural isomorphism) to act on morphisms defining a functor $F$, F: D\to C$, which moreover is left adjoint to the original functor $G$.
This (Once you have the initial morphisms, this extension is really unique, i.e. not just unique up to natural isomorphism.)
Note that this theorem does not involve having an "initial guess" for the left adjoint, but actually constructs it for you from the limited data of the initial morphisms. It is not hard to prove, but not trivial, and is explained relatively well here.
Once you know this, you can really take confidence in "follow your nose"-style adjoint construction. It doesn't even involve having an initial guess for the left adjoint (as a functor), since the theorem constructs it for you from the l
As an example of how this can be useful, think of the inclusion functor $U$ from $AbGrp$ to $Grp$. It's easy to see that any group $H$ has an abelianization $Ab(H) = H/[H,H] \H/[H,H]$ in $AbGrp$ with a map $H\to Ab(H)$ satisfying an initial (universal) property. But then by the above theorem, we know can automatically extend this association automatically extends in a unique way to to act on morphisms as well, defining an abelianization functor $Ab$ which is left adjoint to the inclusion $U$.
This same trick expedites the construction of adjoints in pretty much any situation you can think of. I think it's really worthwhile to sift through the three different characterizations of adjoints given on Wikipedia.
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edited Nov 17 2009 at 7:28 |
Yes, and it is easy to check! In fact, the following theorem makes precise why "this looks like it might work" is so often successful:
Theorem: A functor $G: C\to D$ is a right adjoint functor (i.e. has a left adjoint) if and only if for each object $Y$ in $D$, there exists an initial morphism $\phi_Y:Y\to G(T_Y)$ from $Y$ to $G$. Moreover, once you find such an initial morphism from each $Y$ to $G$, the association $Y\mapsto T_Y$ extends in a unique way (really unique, i.e. not just up to natural isomorphism) to act on morphisms defining a functor $F$, which moreover is left adjoint to the original functor $G$.
This is not hard to prove, but not trivial, and is explained relatively well here. Once you know this, you can really take confidence in "follow your nose"-style adjoint construction.
As an example to illustrate of how helpful this can be useful, think of the inclusion functor $U$ from AbGrp $AbGrp$ to Grp. $Grp$. It's easy to see that any group $H$ has an abelianization $Ab(H) = H/[H,H] \in AbGrp$ satisfying an initial (universal) property, so . But then by the above theorem, we know this association automatically extends in a unique way to to act on morphisms as well, defining an abelianization functor $Ab$ which is left adjoint to the inclusion $U$.
This same trick expedites the construction of adjoints in pretty much any situation you can think of. I think it's really worthwhile to sift through the three different characterizations of adjoints given on Wikipedia.
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edited Nov 17 2009 at 7:12 |
Yes, and it is easy to check! In fact, the following theorem makes precise why "this looks like it might work" is so often successful:
Theorem: A functor $G: C\to D$ is a right adjoint functor (i.e. has a left adjoint) if and only if for each object $Y$ in $D$, there exists an initial morphism $\phi_Y:Y\to G(T_Y))$ G(T_Y)$ from $Y$ to $G$. Moreover, if once you can find such an initial morphism from each $Y$ to $G$, then the association $Y\mapsto T_Y$ extends to act on morphisms in a natural (unique) unique way to define act on morphisms defining a functor $F$, which moreover is left adjoint to the original functor $G$.
This is not hard to prove, but not trivial, and is explained relatively well here. Once you know this, you can really take confidence in "follow your nose"-style adjoint construction.
As an example to illustrate how helpful this can be, think of the inclusion functor $U$ from AbGrp to Grp. It's easy to see that any group $H$ has an abelianization $Ab(H) = H/[H,H] \in AbGrp$ satisfying an initial (universal) property, so by the above theorem, we know this association automatically extends in a unique way to to act on morphisms as well, defining an abelianization functor $Ab$ which is left adjoint to the inclusion $U$.
This same trick expedites the construction of adjoints in pretty much any situation you can think of. I think it's really worthwhile to sift through the three different characterizations of adjoints given on Wikipedia.
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edited Nov 17 2009 at 7:07 |
Yes. I find , and it is easy to check! In fact, the following elementary but frequently overlooked fact to be very useful in practicetheorem makes precise why "this looks like it might work" is so often successful:
Theorem: A functor $G: C\to D$ is a right adjoint functor (i.e. has a left adjoint) if and only if for each object $Y$ in $D$, there exists an initial morphism $\phi_Y:Y\to G(T_Y))$ from $Y$ to $G$. Moreover, if you can find an initial morphism from each $G$ Y$ to each $Y$, G$, then the association $X\mapsto T_X$ Y\mapsto T_Y$ extends to act on morphisms in a natural (unique) way to define a functor $F$, which moreover is right left adjoint to the original functor $F$. G$.
This is not hard to prove, but not trivial, and is explained relatively well here. Once you know this, you can really take confidence in "follow your nose"-style adjoint construction.
As an example to illustrate how helpful this can be, think of the inclusion functor $U$ from AbGrp to Grp. It's easy to see that any group $H$ has an abelianization $Ab(H) = H/[H,H] \in AbGrp$ satisfying an initial (universal) property, so by the above theorem, we know this association automatically extends in a unique way to to act on morphisms as well, defining an abelianization functor $Ab$ which is left adjoint to the inclusion $U$.
This same trick expedites the construction of adjoints in pretty much any situation you can think of. I think it's really worthwhile to sift through the three different characterizations of adjoints given on Wikipedia.
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edited Nov 17 2009 at 6:59 |
Yes. I find the following elementary but frequently overlooked fact to be very useful in practice:
Theorem: A functor $G: C\to D$ is a right adjoint functor (i.e. has a left adjoint) if and only if for each object $Y$ in $D$, there exists an initial morphism from $Y$ to $G$.
In factMoreover, if you can find an initial morphism from $G$ to each $Y$, then one can prove that the association $X\mapsto T_X$ extends to act on morphisms in a unique, natural (unique) way to define a functor $F$, which moreover is right adjoint to the original functor $F$.
This is elementarynot hard to prove, but not trivial, and is explained relatively well here.
As an example to illustrate how helpful this can be, think of the inclusion functor $U$ from AbGrp to Grp. It's easy to see that any group $H$ has an abelianization $Ab(H) = H/[H,H] \in AbGrp$ satisfying an initial (universal) property, so by the above theorem, we know this association automatically extends in a unique way to to act on morphisms as well, defining an abelianization functor $Ab$ which is left adjoint to the inclusion $U$.
This same trick expedites the construction of adjoints in pretty much any situation you can think of. I think it's really worthwhile to sift through the three different characterizations of adjoints given on Wikipedia.
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answered Nov 17 2009 at 6:50 |
Yes. Theorem: A functor $G: C\to D$ is a right adjoint functor (i.e. has a left adjoint) if and only if for each object $Y$ in $D$, there exists an initial morphism from $Y$ to $G$.
In fact, if you can find an initial morphism from $G$ to each $Y$, then one can prove that the association $X\mapsto T_X$ extends to act on morphisms in a unique, natural way to define a functor $F$, which moreover is right adjoint to the original functor $F$. This is elementary, but not trivial, and is explained relatively well here.
As an example to illustrate how helpful this can be, think of the inclusion functor $U$ from AbGrp to Grp. It's easy to see that any group $H$ has an abelianization $Ab(H) = H/[H,H] \in AbGrp$ satisfying an initial (universal) property, so by the above theorem, we know this association automatically in a unique way to to act on morphisms as well, defining an abelianization functor $Ab$ which is left adjoint to the inclusion $U$.
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