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  1. The answer appears to be no; this was discussed on MO previouslyon MO previously.

  2. The symmetric algebra functor is left adjoint to the forgetful functor from commutative algebras to vector spaces (over a field of characteristic zero for simplicity). The exterior algebra functor is left adjoint to the forgetful functor from supercommutative algebras to super vector spaces when restricted to odd vector spaces.

  3. I'm not sure what you mean by this. Surely you don't just mean quotients but universal quotients, e.g. an appropriate natural transformation...?

  1. The answer appears to be no; this was discussed on MO previously.

  2. The symmetric algebra functor is left adjoint to the forgetful functor from commutative algebras to vector spaces (over a field of characteristic zero for simplicity). The exterior algebra functor is left adjoint to the forgetful functor from supercommutative algebras to super vector spaces when restricted to odd vector spaces.

  3. I'm not sure what you mean by this. Surely you don't just mean quotients but universal quotients, e.g. an appropriate natural transformation...?

  1. The answer appears to be no; this was discussed on MO previously.

  2. The symmetric algebra functor is left adjoint to the forgetful functor from commutative algebras to vector spaces (over a field of characteristic zero for simplicity). The exterior algebra functor is left adjoint to the forgetful functor from supercommutative algebras to super vector spaces when restricted to odd vector spaces.

  3. I'm not sure what you mean by this. Surely you don't just mean quotients but universal quotients, e.g. an appropriate natural transformation...?

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Qiaochu Yuan
Qiaochu Yuan
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  1. The symmetric algebra functor is left adjoint to the forgetful functor from commutative algebras to vector spaces (over a field of characteristic zero for simplicity). The exterior algebra functor is left adjoint to the forgetful functor from supercommutative algebras to super vector spaces when restricted to odd vector spaces.

  2. I'm not sure what you mean by this. Surely you don't just mean quotients but universal quotients, e.g. an appropriate natural transformation...?

  1. The answer appears to be no; this was discussed on MO previously.

  2. The symmetric algebra functor is left adjoint to the forgetful functor from commutative algebras to vector spaces (over a field of characteristic zero for simplicity). The exterior algebra functor is left adjoint to the forgetful functor from supercommutative algebras to super vector spaces when restricted to odd vector spaces.

  3. I'm not sure what you mean by this. Surely you don't just mean quotients but universal quotients, e.g. an appropriate natural transformation...?

  1. The symmetric algebra functor is left adjoint to the forgetful functor from commutative algebras to vector spaces (over a field of characteristic zero for simplicity). The exterior algebra functor is left adjoint to the forgetful functor from supercommutative algebras to super vector spaces when restricted to odd vector spaces.

  2. I'm not sure what you mean by this. Surely you don't just mean quotients but universal quotients, e.g. an appropriate natural transformation...?

  1. The answer appears to be no; this was discussed on MO previously.

  2. The symmetric algebra functor is left adjoint to the forgetful functor from commutative algebras to vector spaces (over a field of characteristic zero for simplicity). The exterior algebra functor is left adjoint to the forgetful functor from supercommutative algebras to super vector spaces when restricted to odd vector spaces.

  3. I'm not sure what you mean by this. Surely you don't just mean quotients but universal quotients, e.g. an appropriate natural transformation...?

Source Link
Qiaochu Yuan
Qiaochu Yuan
  • 118.2k
  • 40
  • 447
  • 741

  1. The symmetric algebra functor is left adjoint to the forgetful functor from commutative algebras to vector spaces (over a field of characteristic zero for simplicity). The exterior algebra functor is left adjoint to the forgetful functor from supercommutative algebras to super vector spaces when restricted to odd vector spaces.

  2. I'm not sure what you mean by this. Surely you don't just mean quotients but universal quotients, e.g. an appropriate natural transformation...?