On the Cartesian closedness of [0,1]-Cat and some of its subcategories

TL;DR

This paper characterizes the conditions under which the category of [0,1]-enriched categories is cartesian closed, linking it to various completeness properties and subcategories.

Contribution

It provides a complete description of left continuous triangular norms ensuring cartesian closedness of [0,1]-Cat and relates this property to several important subcategories.

Findings

Identifies all left continuous triangular norms for cartesian closedness.

Shows equivalence of cartesian closedness between [0,1]-Cat and its key subcategories.

Connects cartesian closedness with completeness properties in [0,1]-categories.

Abstract

We describe all left continuous triangular norms for which the category [0,1]-Cat of real-enriched categories and functors is cartesian closed. We furthermore show that the cartesian closedness of [0,1]-Cat is equivalent to the cartesian closedness of either (and thus all) of the following subcategories: the full subcategory of Cauchy complete [0,1]-categories; the subcategory of Yoneda complete [0,1]-categories and Yoneda continuous [0,1]-functors; the full subcategory of Smyth complete [0,1]-categories; and the full subcategory of finite [0,1]-categories.