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

Abstract

Let $\&$ be a continuous triangular norm on the unit interval $[0,1]$ and $\mathbf{A}$ be a cartesian closed and stable subconstruct of the category consisting of all real-enriched categories. Firstly, it is shown that the category $\mathbf{A}$ 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 form a cartesian closed category.