nLab Field

Definition

$Field$ 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 $Field$ not being a very well-behaved category.

Properties

• $Field$ 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, $Field$ is a locally multipresentable category, which means in particular that it has connected limits and multicolimits.
• The forgetful functor $U\colon Field\to Set$ does not have a left adjoint (hence $Field$ is not a reflective subcategory of $Set$), hence there is no free field construction (contrary to many free functors for other algebraic categories). The same holds for weaker forgetful functors like $U\colon Field\to Ring$, $U\colon Field\to Ab$ or the group of units $-^\times\colon Field\to Ab$. (Riehl 17, Example 4.1.11.)
• Every morphism in $Field$ is a monomorphism, hence $Field$ is a left-cancellative category.
• The isomorphisms in $Field$ are the bijective homomorphisms. (Riehl 17, Examples 1.1.6. (ii))

Subcategories

$Field$ is not connected as there are no field homomorphisms between fields of different characteristic. The connected component (full subcategory of $Field$) corresponding to characteristic $p$ (with $p=0$ or $p$ prime) is denoted $Field_p$.

The field of rational numbers $\mathbb{Q}$ is the initial object of $Field_0$ and the prime field $\mathbb{F}_p$ is the initial object of $Field_p$, but none are in $Field$, which has neither an initial nor terminal object. (Riehl 17, Examples 1.6.18. (vi))

References

• Emily Riehl, Category Theory in Context, Dover Publications (2017) [pdf]