# A natural fibration for rings

### Alessandro Andrea Bosi

Università di Roma Tre, Italy### Alberto Facchini

Università di Padova, Italy

## 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 $F\colon L\to\mathsf{Ring}$ 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 $\mathcal{H}\colon\mathsf{Ring}\to\mathsf{RingedParOrd}_0$ 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 $Z\colon\mathsf{RingedParOrd}_0\to\mathsf{Ring}$. The category $\mathsf{RingedParOrd}_0$ is a fibred category.

## Cite this article

Alessandro Andrea Bosi, Alberto Facchini, A natural fibration for rings. Rend. Sem. Mat. Univ. Padova 145 (2021), pp. 167–180

DOI 10.4171/RSMUP/76