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I'm embarassed because I'm surely missing something obvious... the very first exercise in Categories for the Working Mathematician is:
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 map f of manifolds gives rise to a map f* of 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?
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 map f of manifolds gives rise to a map f* of 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?