A natural fibration for rings

Abstract

A ringed partially ordered set with zero is a pair $(L,F)$, where $L$ is a partially ordered set with a least element $0_L$ and $F:L\to \mathsf{Ring}$ is a covariant functor. Here the partially ordered set $L$ is given a category structure in the usual way and $\mathsf{Ring}$ denotes the category of associative rings with identity. Let ${\mathsf{RingedParOrd}}_0$ be the category of ringed partially ordered sets with zero. There is a functor $\mathcal{H}:\mathsf{Ring}\to {\mathsf{RingedParOrd}}_0$ that associates to any ring $R$ a ringed partially ordered set with zero $(\mathrm{Hom}(R),F_R)$. The functor $\mathcal{H}$ has a left inverse $Z:{\mathsf{RingedParOrd}}_0\to \mathsf{Ring}$. The category ${\mathsf{RingedParOrd}}_0$ is a fibred category.