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Double Orthogonal Factorization Systems

C. B. Aberlé, Elena Caviglia, Matthew Kukla, Rubén Maldonado, Luca Mesiti, Dorette Pronk, Tanjona Ralaivaosaona

TL;DR

The paper develops a theory of double orthogonal factorization systems ($\mathrm{DOFS}$) for double categories by pairing OFS on arrows with compatible OFS on double cells and studies their behavior under monadic constructions and double fibrations. It provides an internal, cell-wise formulation, proves a fundamental monadicity theorem that identifies DOFS as algebras for a generalized 2-monad, and demonstrates how arrow OFS lift to DOFS across a range of examples (spans, relations, bimodules, profunctors). The work establishes extension and restriction mechanisms to propagate an arrow OFS to a DOFS, with initial/terminal liftings among possible extensions, and applies these to concrete double categories such as Mod, Rel, Pan, and Sq. It also analyzes how DOFS interact with double fibrations, offering lifting results that extend base factorizations to total double categories and clarifies the landscape of lax/pseudo/strict colax variants. Overall, the paper provides a scalable framework for image-like factorizations in double categorical settings, enabling robust factorization theory in higher-dimensional contexts with applications to spans, relations, and profunctor-like structures.

Abstract

We define strict and lax orthogonal factorization systems on double categories. These consist of an orthogonal factorization system on arrows and one on double cells that are compatible with each other. Our definitions are motivated by several explicit examples, including factorization systems on double categories of spans, relations and bimodules. We then prove monadicity results for orthogonal factorization systems on double categories in order to justify our definitions. For fibrant double categories we discuss the structure of the double orthogonal factorization systems that have a given orthogonal factorization system on the arrows in common. Finally, we study the interaction of orthogonal factorization systems on double categories with double fibrations.

Double Orthogonal Factorization Systems

TL;DR

The paper develops a theory of double orthogonal factorization systems () for double categories by pairing OFS on arrows with compatible OFS on double cells and studies their behavior under monadic constructions and double fibrations. It provides an internal, cell-wise formulation, proves a fundamental monadicity theorem that identifies DOFS as algebras for a generalized 2-monad, and demonstrates how arrow OFS lift to DOFS across a range of examples (spans, relations, bimodules, profunctors). The work establishes extension and restriction mechanisms to propagate an arrow OFS to a DOFS, with initial/terminal liftings among possible extensions, and applies these to concrete double categories such as Mod, Rel, Pan, and Sq. It also analyzes how DOFS interact with double fibrations, offering lifting results that extend base factorizations to total double categories and clarifies the landscape of lax/pseudo/strict colax variants. Overall, the paper provides a scalable framework for image-like factorizations in double categorical settings, enabling robust factorization theory in higher-dimensional contexts with applications to spans, relations, and profunctor-like structures.

Abstract

We define strict and lax orthogonal factorization systems on double categories. These consist of an orthogonal factorization system on arrows and one on double cells that are compatible with each other. Our definitions are motivated by several explicit examples, including factorization systems on double categories of spans, relations and bimodules. We then prove monadicity results for orthogonal factorization systems on double categories in order to justify our definitions. For fibrant double categories we discuss the structure of the double orthogonal factorization systems that have a given orthogonal factorization system on the arrows in common. Finally, we study the interaction of orthogonal factorization systems on double categories with double fibrations.

Paper Structure

This paper contains 11 sections, 23 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Categories of factorization systems
  3. Double factorization systems
  4. Motivating Examples
  5. Internal definition
  6. Examples of DOFS on double categories constructed from 2-categories
  7. Monadicity of double factorization systems
  8. Extending an arrow OFS to a DOFS
  9. Theoretical results
  10. Examples
  11. Double factorization systems and double fibrations

Key Result

Theorem 1.1

Let $\mathpzc{K}\xspace$ be a 2-category and let $(T\colon \mathpzc{K}\xspace\to \mathpzc{K}\xspace, \eta\colon\operatorname{Id}_{} \ext@arrow 0055{\Rightarrowfill@}{}{}T, \mu\colon T^2 \ext@arrow 0055{\Rightarrowfill@}{}{} T)$ be a 2-monad whose underlying 2-functor preserves pullbacks. Then $T$ in such that where $\mathpzc{Cat}\xspace {{}}() '[o]{ \hbox{-}\mathpzc{Alg}\xspace{{}}[l]{T}}$ (respe

Theorems & Definitions (83)

  • Theorem 1.1: Theorem \ref{['teorgenmonadicity']}
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4: kt93
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7: grandistholen
  • Definition 2.8
  • Remark 2.10
  • ...and 73 more