A natural fibration for rings (Q2039374)

The authors study the functors between several categories, the most important of which are the category Ring of all (associative, unital) rings, the category ParOrd\(_0\) of all partially ordered sets with the least element, and the category RingedParOrd\(_0\) of all ringed partially ordered sets with the least element. A ringed partially ordered set (with the least element) is a pair \((L, F)\), where \(L\) is a partially ordered set (with the least element) and \(F:L\rightarrow\) Ring is a covariant functor.\par Ringed partially ordered sets are similar to ringed spaces. The results of the present paper are versions of the results known for ringed spaces to the case of ringed partially ordered sets.\par The authors construct contravariant functors \({\mathcal H}:\) Ring\(\rightarrow\) RingedParOrd\(_0\) and \(Z:\) RingedParOrd\(_0\rightarrow\) Ring such that \(Z\circ{\mathcal H}=1_{\mathrm{Ring}}\), but \(\mathcal H\) and \(Z\) are not adjoint on the right and not adjoint on the left.\par In the last chapter, the authors construct a fibration \(p:\)RingedParOrd\(_0\rightarrow\) ParOrd\(_0\), which is actually a projection functor.