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On logical parameterizations and functional representability in local set theories

Enrique Ruiz Hernández, Pedro Solórzano

Abstract

There is a well-known inclusion $ι_\mathscr{E}$ of a topos $\mathscr{E}$ in the linguistic topos $\mathscr{T}(Σ)$ of its internal language $Σ$ that proves both toposes to be equivalent. There is also a canonical translation $η_S$ for any local set theory $S$ into the local set theory $Σ$ of its linguistic topos. Starting from a local set theory, this yields two a priori distinct inclusions from $\mathscr{T}(S)$ to $\mathscr{T}(Σ)$. Herein, these two functors are proved to be isomorphic. Furthermore, the concept of logical parameterization is investigated and then applied to see that $ι_{\mathscr{T}(S)}$ parameterizes $\mathscr{T}(η_S)$ in such a way that syntactic $S$-functions are represented by themselves in $Σ$.

On logical parameterizations and functional representability in local set theories

Abstract

There is a well-known inclusion of a topos in the linguistic topos of its internal language that proves both toposes to be equivalent. There is also a canonical translation for any local set theory into the local set theory of its linguistic topos. Starting from a local set theory, this yields two a priori distinct inclusions from to . Herein, these two functors are proved to be isomorphic. Furthermore, the concept of logical parameterization is investigated and then applied to see that parameterizes in such a way that syntactic -functions are represented by themselves in .

Paper Structure

This paper contains 15 sections, 24 theorems, 158 equations.

Table of Contents

  1. Preliminaries
  2. Toposes
  3. Local languages
  4. Local set theories
  5. Representability
  6. Interpretations
  7. Internal language and local set theory of a topos
  8. Translations
  9. Parameterizations
  10. Preimage translations
  11. Universal parameterizations
  12. Functional representability
  13. Canonical parameterization
  14. Strict representation
  15. A remark on the syntactic sets of $\mathrm{Th}(\mathscr{T}(S))$.

Key Result

Proposition 1.3.1

Let $X,Y,\lvert f\rvert$ be $S$-sets with $\vdash_S\lvert f\rvert\subseteq X\times Y$. Then, $x\in X\vdash_S\exists!y\in Y(\langle x,y\rangle\in\lvert f\rvert)$ if and only if $x\in X\vdash_S\exists y\in Y(\langle x,y\rangle\in\lvert f\rvert)$ and $\langle x,y\rangle\in\lvert f\rvert,\langle x,y'\ra

Theorems & Definitions (49)

  • Proposition 1.3.1
  • proof
  • Proposition 1.4.1: MR972257, 3.16.4
  • Lemma 1.5.3
  • proof
  • Remark 1.5.4
  • Theorem 1.6.3: MR972257 3.36
  • Theorem 1.6.4: MR972257 3.37
  • Remark 2.1.2
  • Definition 2.1.3
  • ...and 39 more