nLab
Field
Redirected from "category of fields".
Contents
Contents
Definition
is the category with fields as objects and field homomorphisms as morphisms. It is a full subcategory of CRing, the more general category of commutative rings, which is more often considered. This is due to not being a very well-behaved category.
Properties
- is neither finitely complete nor finitely cocomplete, meaning in particular, that it has neither products nor coproducts. (Riehl 17, p. 126) This implies that it is not locally presentable.
- However, is a locally multipresentable category, which means in particular that it has connected limits and multicolimits.
- The forgetful functor does not have a left adjoint (hence is not a reflective subcategory of ), hence there is no free field construction (contrary to many free functors for other algebraic categories). The same holds for weaker forgetful functors like , or the group of units . (Riehl 17, Example 4.1.11.)
- Every morphism in is a monomorphism, hence is a left-cancellative category.
- The isomorphisms in are the bijective homomorphisms. (Riehl 17, Examples 1.1.6. (ii))
Subcategories
is not connected as there are no field homomorphisms between fields of different characteristic. The connected component (full subcategory of ) corresponding to characteristic (with or prime) is denoted .
The field of rational numbers is the initial object of and the prime field is the initial object of , but none are in , which has neither an initial nor terminal object. (Riehl 17, Examples 1.6.18. (vi))
References
Last revised on February 21, 2024 at 16:37:21.
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