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Yuhao Huang
Yuhao Huang
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for forgetful functors, we can usually find their left adjoint as some "free objects", e.g. the forgetful functor: AbGp -> Set, its left adjoint sends a set to the "free ab. gp gen. by it". This happens even in some non-trivial cases(e.g. the forgetful functor from crystals to schemes). So my question is, why these happen? i.e. why that a functor forgets some structure (in certain cases) implies that they have a left adjoint? Thanks.

Edited: Many answers are on the "linear" examples, which I'm sort of comfortable with. But actually what motivates me to ask this question is the fact that Jet-scheme appears as a free crystal, i.e. a left adjoint to the forgetful functor from crystal of schemes to schemes (over some basis), which is surprising to me. I don't want to go to the technical details of this example, but do want to see a reason for such "non-linear" examples. [At this moment I think maybe checking the commutivity with limits is a good way to tell something.]

Feel free to give more comments.

for forgetful functors, we can usually find their left adjoint as some "free objects", e.g. the forgetful functor: AbGp -> Set, its left adjoint sends a set to the "free ab. gp gen. by it". This happens even in some non-trivial cases(e.g. the forgetful functor from crystals to schemes). So my question is, why these happen? i.e. why that a functor forgets some structure (in certain cases) implies that they have a left adjoint? Thanks.

Edited: Many answers are on the "linear" examples, which I'm sort of comfortable with. But actually what motivates me to ask this question is the fact that Jet-scheme appears as a free crystal, i.e. a left adjoint to the forgetful functor from crystal of schemes to schemes (over some basis), which is surprising to me. I don't want to go to the technical details of this example, but do want to see a reason for such "non-linear" examples. [At this moment I think maybe checking the commutivity with limits is a good way to tell something.]

Feel free to give more comments.

for forgetful functors, we can usually find their left adjoint as some "free objects", e.g. the forgetful functor: AbGp -> Set, its left adjoint sends a set to the "free ab. gp gen. by it". This happens even in some non-trivial cases. So my question is, why these happen? i.e. why that a functor forgets some structure (in certain cases) implies that they have a left adjoint? Thanks.

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Source Link
Yuhao Huang
Yuhao Huang
  • 5.1k
  • 1
  • 39
  • 62

for forgetful functors, we can usually find their left adjoint as some "free objects", e.g. the forgetful functor: AbGp -> Set, its left adjoint sends a set to the "free ab. gp gen. by it". This happens even in some non-trivial cases(e.g. the forgetful functor from crystals to schemes). So my question is, why these happen? i.e. why that a functor forgets some structure (in certain cases) implies that they have a left adjoint? Thanks.

Edited: Many answers are on the "linear" examples, which I'm sort of comfortable with. But actually what motivates me to ask this question is the fact that Jet-scheme appears as a free crystal, i.e. a left adjoint to the forgetful functor from crystal of schemes to schemes (over some basis), which is surprising to me. I don't want to go to the technical details of this example, but do want to see a reason for such "non-linear" examples. [At this moment I think maybe checking the commutivity with limits is a good way to tell something.]

Feel free to give more comments.

for forgetful functors, we can usually find their left adjoint as some "free objects", e.g. the forgetful functor: AbGp -> Set, its left adjoint sends a set to the "free ab. gp gen. by it". This happens even in some non-trivial cases(e.g. the forgetful functor from crystals to schemes). So my question is, why these happen? i.e. why that a functor forgets some structure (in certain cases) implies that they have a left adjoint? Thanks.

for forgetful functors, we can usually find their left adjoint as some "free objects", e.g. the forgetful functor: AbGp -> Set, its left adjoint sends a set to the "free ab. gp gen. by it". This happens even in some non-trivial cases(e.g. the forgetful functor from crystals to schemes). So my question is, why these happen? i.e. why that a functor forgets some structure (in certain cases) implies that they have a left adjoint? Thanks.

Edited: Many answers are on the "linear" examples, which I'm sort of comfortable with. But actually what motivates me to ask this question is the fact that Jet-scheme appears as a free crystal, i.e. a left adjoint to the forgetful functor from crystal of schemes to schemes (over some basis), which is surprising to me. I don't want to go to the technical details of this example, but do want to see a reason for such "non-linear" examples. [At this moment I think maybe checking the commutivity with limits is a good way to tell something.]

Feel free to give more comments.

Source Link
Yuhao Huang
Yuhao Huang
  • 5.1k
  • 1
  • 39
  • 62

Why forgetful functors usually have LEFT adjoint?

for forgetful functors, we can usually find their left adjoint as some "free objects", e.g. the forgetful functor: AbGp -> Set, its left adjoint sends a set to the "free ab. gp gen. by it". This happens even in some non-trivial cases(e.g. the forgetful functor from crystals to schemes). So my question is, why these happen? i.e. why that a functor forgets some structure (in certain cases) implies that they have a left adjoint? Thanks.