+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Category theory +-- {: .hide} [[!include category theory - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Definition __$Field$__ is the [[category]] with [[field|fields]] as [[objects]] and [[field homomorphism|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 category|finitely complete]] nor [[finitely cocomplete category|finitely cocomplete]], meaning in particular, that it has neither [[products]] nor [[coproducts]]. ([Riehl 17, p. 126](#RiehlCTInContext)) 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.](#RiehlCTInContext)) * 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)](#RiehlCTInContext)) ## Subcategories $Field$ is not [[connected category|connected]] as there are no [[field homomorphism|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)](#RiehlCTInContext)) ## References * {#RiehlCTInContext} [[Emily Riehl]], _[[Category Theory in Context]]_, Dover Publications (2017) [[pdf](https://emilyriehl.github.io/files/context.pdf)] [[!redirects category of fields]] [[!redirects categories of fields]]