The change-base issue for \(\Omega\)-categories
If \((\Omega ,*,I)\) is a commutative and unital quantale (that is, the unit element \(I\) of \(\Omega \) coincides with the top element in \(\Omega \)), an \( \Omega\)-category is a set \(A\) together with an assignment of an element \( A(a,b)\in \Omega \) to every ordered pair \((a,b)\in A\times A\) such that \( I\leq A(a,a)\) for every \(a\in A\) and \(A(a,b)*A(b,c)\leq A(a,c)\) for all \( a,b,c\in A.\) An \(\Omega\)-functor between \(\Omega\)-categories \(A\) and \(B\) is a function \(f:A\rightarrow B\) such that \(A(a,b)\leq B(f(a),f(b)),\) for all \(a,b\in A.\) If \(G:(\Omega ,*,I)\rightarrow (\Omega ^{\prime },*^{\prime },I^{\prime })\) is a closed unital map between commutative, unital quantales, then \(G\) induces a canonical functor \(\overline{G}:\Omega \)-\({\mathcal C}at\rightarrow \Omega ^{\prime }\)-\({\mathcal C}at\). This paper is concerned with some basic questions about \( \overline{G}:\) when does \(\overline{G}\) have a left adjoint which is also of this form; when \(\overline{G}\) is an equivalence of categories and when does \(\overline{G}\) preserve completeness? The main results are: (1) when \(\Omega\), \(\Omega ^{\prime }\) are integral, \( G:\Omega \rightarrow \Omega ^{\prime }\) and \(F:\Omega ^{\prime }\rightarrow \Omega \) are closed unital maps, then \(\overline{F}\) is a left adjoint of \( \overline{G}\) if and only if \(F\) is a left adjoint of \(G;\) (2) \(\overline{G}\) is an equivalence of categories if and only if \(G\) is an isomorphism in the category of commutative unital quantales and closed unital maps; and (3) a sufficient condition is obtained for \(\overline{G}\) to preserve completeness in the sense that \(\overline{G}A\) is a complete \(\Omega ^{\prime }\)-category whenever \(A\) is a complete \(\Omega \)-category.
scientific article; zbMATH DE number 5322654
scientific article