Cartesian closed and stable subconstructs of [0,1]-Cat

Abstract

Let & be a continuous triangular norm on the unit interval $[0,1]$ and $[0,1]$-Cat be the category of all real-enriched categories. Firstly, it is shown that a stable subconstruct A of $[0,1]$-Cat is cartesian closed if and only if it is determined by a suitable subset $S\subseteq M^2$ of ${[0,1]}^2$, where M is the set of all elements x in $[0,1]$ such that x&x is idempotent. Secondly, it is shown that all Yoneda complete real-enriched categories valued in the set M and Yoneda continuous $[0,1]$-functors also form a cartesian closed category.