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Cartesian closedness of the category of real-valued sets, I

Lili Shen, Jian Zhang

TL;DR

The paper characterizes when the category of real-valued sets $[0,1]_*$-Set, for a continuous t-norm $*$ on $[0,1]$, is cartesian closed. By developing an elementary treatment of $[0,1]_*$-sets and employing Cauchy completion, it reduces the problem to the min-t-norm case, showing Cartesian closedness holds precisely when $*$ is the minimum t-norm. The argument leverages a canonical equivalence with separated Cauchy complete sets and a counterexample construction for non-min t-norms to establish the impossibility of exponentials in those cases. As a corollary, the corresponding $Q$-Set results imply that a topos-like structure arises only in the min-t-norm scenario, aligning with existing topos theory in this specialized setting.

Abstract

Let $[0,1]_*$ be the unit interval $[0,1]$ equipped with a continuous t-norm $*$. It is shown that the category of $[0,1]_*$-sets is cartesian closed if, and only if, $*$ is the minimum t-norm on $[0,1]$.

Cartesian closedness of the category of real-valued sets, I

TL;DR

The paper characterizes when the category of real-valued sets -Set, for a continuous t-norm on , is cartesian closed. By developing an elementary treatment of -sets and employing Cauchy completion, it reduces the problem to the min-t-norm case, showing Cartesian closedness holds precisely when is the minimum t-norm. The argument leverages a canonical equivalence with separated Cauchy complete sets and a counterexample construction for non-min t-norms to establish the impossibility of exponentials in those cases. As a corollary, the corresponding -Set results imply that a topos-like structure arises only in the min-t-norm scenario, aligning with existing topos theory in this specialized setting.

Abstract

Let be the unit interval equipped with a continuous t-norm . It is shown that the category of -sets is cartesian closed if, and only if, is the minimum t-norm on .
Paper Structure (4 sections, 12 theorems, 123 equations)

This paper contains 4 sections, 12 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Real-valued sets
  3. Cauchy complete real-valued sets
  4. Cartesian closedness of the category of real-valued sets

Key Result

Proposition 2.2

(See Pu2012.) The following identities hold in every continuous t-norm $[a,b]_*$:

Theorems & Definitions (26)

  • Example 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Remark 2.8
  • Proposition 3.1
  • ...and 16 more