I'm embarassed because I'm surely missing something obvious... the very first exercise in Categories for the Working Mathematician is:(adsbygoogle = window.adsbygoogle || []).push({});

Show how each of the following constructions can be regarded as a functor: The field of quotients of an integral domain; the Lie algebra of a Lie group.

The latter one is easy, since any mapfof manifolds gives rise to a mapfof the tangent spaces, and as I recall, it all works out properly._{*}

But the former is bothering me. Assuming commutative domains, taking quotients cannot be a map Domain-->Field, and considering the map [itex]\mathbb{Z} \rightarrow \mathbb{Z}_5[/itex]:

Taking the quotient field cannot be covariant because that would yield a map of fields [itex]\mathbb{Q} \rightarrow \mathbb{Z}_5[/itex], and it cannot be contravariant because that would yield a map of fields [itex]\mathbb{Z}_5 \rightarrow \mathbb{Q}[/itex].

So in what way have I confused myself? Or is this actually a bad exercise?

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Quotient field vs Functor

Loading...

Similar Threads - Quotient field Functor | Date |
---|---|

I Quotient Rings ... Remarks by Adkins and Weintraub ... | Saturday at 7:36 PM |

I Simple Modules and quotients of maximal modules, Bland Ex 13 | Feb 3, 2017 |

I Why only normal subgroup is used to obtain group quotient | Mar 5, 2016 |

Quotient field of the integral closure of a ring | Dec 11, 2014 |

Some questions of field of quotients | Jun 5, 2010 |

**Physics Forums - The Fusion of Science and Community**