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Rado
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In the introduction of his class field theory notes Milne mentions that some famous mathematicians failed to ask if the Artin isomorphism is canonical (between Gal(L/K)$Gal(L/K)$ and C_m/H$C_m/H$ where H$H$ is generated by the split primes in L$L$). Does this mean:

1)in category theory terms: there is a natural transformation between the functors from abelian extensions over K to abelian groups given by Gal(/K) and C_m/H$Gal(?/K)$ and $C_m/H?$ (where H_H? is generated by the primes split over _/K$?/K$).

2)or some kind of vaguer statement about whether we need to make choices along the definition of the map.

or maybe 2) is precisely encoded in the definition of 1).

In the introduction of his class field theory notes Milne mentions that some famous mathematicians failed to ask if the Artin isomorphism is canonical (between Gal(L/K) and C_m/H where H is generated by the split primes in L). Does this mean:

1)in category theory terms: there is a natural transformation between the functors from abelian extensions over K to abelian groups given by Gal(/K) and C_m/H (where H_ is generated by the primes split over _/K).

2)or some kind of vaguer statement about whether we need to make choices along the definition of the map.

or maybe 2) is precisely encoded in the definition of 1).

In the introduction of his class field theory notes Milne mentions that some famous mathematicians failed to ask if the Artin isomorphism is canonical (between $Gal(L/K)$ and $C_m/H$ where $H$ is generated by the split primes in $L$). Does this mean:

1)in category theory terms: there is a natural transformation between the functors from abelian extensions over K to abelian groups given by $Gal(?/K)$ and $C_m/H?$ (where H? is generated by the primes split over $?/K$).

2)or some kind of vaguer statement about whether we need to make choices along the definition of the map.

or maybe 2) is precisely encoded in the definition of 1).

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Rado
Rado
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In the introduction of his class field theory notes Milne mentions that some famous mathematicians failed to ask if the Artin isomorphism is canonical (between Gal(L/K) and C_m/H where H is generated by the split primes in L). Does this mean:

1)in category theory terms: there is a natural transformation between the functors from abelian extensions over K to abelian groups given by Gal(/K) and C_m/H (where H_ is generated by the primes split over _/K). 2

2)or some kind of vaguer statement about whether we need to make choices along the definition of the map.

or maybe 2) is precisely encoded in the definition of 1).

In the introduction of his class field theory notes Milne mentions that some famous mathematicians failed to ask if the Artin isomorphism is canonical (between Gal(L/K) and C_m/H where H is generated by the split primes in L). Does this mean:

1)in category theory terms: there is a natural transformation between the functors from abelian extensions over K to abelian groups given by Gal(/K) and C_m/H (where H_ is generated by the primes split over _/K). 2)or some kind of vaguer statement about whether we need to make choices along the definition of the map.

or maybe 2) is precisely encoded in the definition of 1).

In the introduction of his class field theory notes Milne mentions that some famous mathematicians failed to ask if the Artin isomorphism is canonical (between Gal(L/K) and C_m/H where H is generated by the split primes in L). Does this mean:

1)in category theory terms: there is a natural transformation between the functors from abelian extensions over K to abelian groups given by Gal(/K) and C_m/H (where H_ is generated by the primes split over _/K).

2)or some kind of vaguer statement about whether we need to make choices along the definition of the map.

or maybe 2) is precisely encoded in the definition of 1).

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Rado
Rado
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remark in milne's class field theory notes

In the introduction of his class field theory notes Milne mentions that some famous mathematicians failed to ask if the Artin isomorphism is canonical (between Gal(L/K) and C_m/H where H is generated by the split primes in L). Does this mean:

1)in category theory terms: there is a natural transformation between the functors from abelian extensions over K to abelian groups given by Gal(/K) and C_m/H (where H_ is generated by the primes split over _/K). 2)or some kind of vaguer statement about whether we need to make choices along the definition of the map.

or maybe 2) is precisely encoded in the definition of 1).