When is the category [0,1]-Cat cartesian closed?
Hongliang Lai, Qingzhu Luo
TL;DR
The paper characterizes exactly which left-continuous triangular norms $\&$ on $[0,1]$ render the category $[0,1]$-Cat cartesian closed, showing that nontrivial norms beyond $\wedge$ exist and are governed by a precise interval-decomposition condition involving idempotents. It proves that this CCC property is equivalent to similar CCC-ness for natural subcategories CauCom, YonCom, and SmyCom, and that under these norms the corresponding exponentials preserve Cauchy, Yoneda, and Smyth completeness. The results extend the known fact that only the wedge operation yields CCC in the continuous case, by identifying a broader family of left-continuous t-norms that still admit exponentials. The findings provide a concrete framework for real-enriched CCC and its complete-subcategory variants, with potential implications for quantitative domain theory and enriched functional spaces.
Abstract
In this paper, we describe all left continuous triangular norms such that the category [0,1]-Cat, consisting of all real-enriched categories, is cartesian closed. Moreover, in this case, we show that, its subcategories CauCom consisting of Cauchy complete objects and YonCom consisting of Yoneda complete objects and Yoneda continuous [0,1]-functors are also cartesian closed.