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Elgot Categories and Abacus Programs

Chad Nester

TL;DR

This work introduces Elgot categories, a sort of distributive monoidal category with additional structure in which the partial recursive functions are representable and constructs an initial Elgot category, the morphisms of which coincide with a lightly modified version of Lambek's abacus programs.

Abstract

We introduce Elgot categories, a sort of distributive monoidal category with additional structure in which the partial recursive functions are representable. Moreover, we construct an initial Elgot category, the morphisms of which coincide with a lightly modified version of Lambek's abacus programs. The partial functions that are strongly representable in this initial Elgot category are precisely the partial recursive ones.

Elgot Categories and Abacus Programs

TL;DR

This work introduces Elgot categories, a sort of distributive monoidal category with additional structure in which the partial recursive functions are representable and constructs an initial Elgot category, the morphisms of which coincide with a lightly modified version of Lambek's abacus programs.

Abstract

We introduce Elgot categories, a sort of distributive monoidal category with additional structure in which the partial recursive functions are representable. Moreover, we construct an initial Elgot category, the morphisms of which coincide with a lightly modified version of Lambek's abacus programs. The partial functions that are strongly representable in this initial Elgot category are precisely the partial recursive ones.

Paper Structure

This paper contains 10 sections, 21 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Related Work
  3. Synopsis
  4. Traced and Distributive Monoidal Categories
  5. Categorical Representability of Partial Recursive Functions
  6. Elgot Categories
  7. Abacus Programs
  8. Conclusions Future Work
  9. Distributive Monoidal Categories
  10. Proof of Theorem \ref{['thm:abacus-elgot']}

Key Result

lemma 1

Let $((\mathbb{C},+,0,\sigma^+),(\mu,\eta))$ be a cocartesian monoidal category, and let $(\mathbb{C},\otimes,I)$ be a monoidal category. Suppose that $\lambda^\bullet$, $\rho^\bullet$, $\delta^l$, and $\delta^r$ are indexed families of morphisms as in Definition def:distributive-monoidal-category.

Theorems & Definitions (51)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • lemma 1: after Labella2003
  • definition 5
  • definition 6
  • definition 7
  • proof
  • definition 8
  • ...and 41 more