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Partial Linearity in Categories

Roy Ferguson, Zurab Janelidze

TL;DR

This work addresses partial linearity in categories by replacing the traditional isomorphism between coproducts and products with isomorphisms between designated monoidal structures, realized through a (potential) natural transformer $i: \oplus \to \otimes$. It develops a coherence framework via linearisers, introducing prelinear and lineariser concepts, and proves a coherence theorem ensuring canonical maps between $n$-fold sums and products are constrained to identity-like presentations. It also generalizes central morphisms to the partial linearity setting, showing central morphisms form monoids under addition with distributive composition, and provides criteria linking these monoids to full linearity of the category. The results offer a modular, monoidal approach to partial linearity with potential implications for categorical algebra and related mathematical structures.

Abstract

In this paper we study partial linearity in a category by replacing isomorphism between coproducts and products in a linear category with isomorphism between suitable monoidal structures on a category. The main results a coherence theorem and a generalization of the theory of central morphisms from unital categories to our context of partial linearity

Partial Linearity in Categories

TL;DR

This work addresses partial linearity in categories by replacing the traditional isomorphism between coproducts and products with isomorphisms between designated monoidal structures, realized through a (potential) natural transformer . It develops a coherence framework via linearisers, introducing prelinear and lineariser concepts, and proves a coherence theorem ensuring canonical maps between -fold sums and products are constrained to identity-like presentations. It also generalizes central morphisms to the partial linearity setting, showing central morphisms form monoids under addition with distributive composition, and provides criteria linking these monoids to full linearity of the category. The results offer a modular, monoidal approach to partial linearity with potential implications for categorical algebra and related mathematical structures.

Abstract

In this paper we study partial linearity in a category by replacing isomorphism between coproducts and products in a linear category with isomorphism between suitable monoidal structures on a category. The main results a coherence theorem and a generalization of the theory of central morphisms from unital categories to our context of partial linearity
Paper Structure (5 sections, 12 theorems, 10 equations)

This paper contains 5 sections, 12 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linearisers
  4. Centrality with respect to sum and product structures
  5. Acknowledgement

Key Result

Proposition 1

Suppose $\mathbb{C}$ is a category equipped with a sum structure $(\oplus,0, \alpha_\oplus,\lambda_\oplus,\rho_\oplus)$ and a product structure $(\otimes,1,\alpha_\otimes,\lambda_\otimes,\rho_\otimes)$ where there exists a natural transformation $i \colon \oplus \to \otimes$. Then $\mathbb{C}$ is po

Theorems & Definitions (20)

  • Proposition 1
  • proof
  • Definition 2
  • Remark 3
  • Lemma 4
  • Theorem 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Lemma 9
  • ...and 10 more