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Rectangular torsion theories

Elena Caviglia, Zurab Janelidze, Luca Mesiti

TL;DR

This work defines rectangular torsion theories as pointed-structure pretorsion theories where the canonical functor $\mathcal{C}\to\mathcal{T}\times\mathcal{F}$ is an equivalence, connecting them to internal rectangular bands in the 2-category of pointed categories. It establishes a 2-equivalence between the 2-category of internal rectangular bands and rectangular torsion theories, and shows these theories decompose like products of $(\mathcal{T},\mathcal{T},\mathbf{0})$ and $(\mathbf{0},\mathbf{0},\mathcal{F})$, mirroring rectangular bands in semigroup theory. The pointed case is given a monadic description: rectangular torsion theories correspond to pseudo-algebras for a 2-monad $M$ on bi-quasi-pointed categories, yielding a 2-equivalence with the category of $M$-pseudo-algebras and enabling bilimit constructions. A concrete criterion for when a class of epimorphisms yields a rectangular torsion theory is provided, linking to normal projections, with implications for categories like vector spaces and pointed objects in a topos. Overall, the paper advances a 2-categorical and monadic framework to recognize and construct rectangular torsion theories and their bilimits, offering practical criteria and canonical decompositions.

Abstract

In this paper we introduce and study \emph{rectangular torsion theories}, i.e.\ those torsion theories $(\C,\T,\F)$ with $\C$ a pointed category, where the canonical functor $\C\to \T\times\F$ is an equivalence of categories. In particular, we show that these are precisely the internal rectangular bands in the 2-category of pointed categories.

Rectangular torsion theories

TL;DR

This work defines rectangular torsion theories as pointed-structure pretorsion theories where the canonical functor is an equivalence, connecting them to internal rectangular bands in the 2-category of pointed categories. It establishes a 2-equivalence between the 2-category of internal rectangular bands and rectangular torsion theories, and shows these theories decompose like products of and , mirroring rectangular bands in semigroup theory. The pointed case is given a monadic description: rectangular torsion theories correspond to pseudo-algebras for a 2-monad on bi-quasi-pointed categories, yielding a 2-equivalence with the category of -pseudo-algebras and enabling bilimit constructions. A concrete criterion for when a class of epimorphisms yields a rectangular torsion theory is provided, linking to normal projections, with implications for categories like vector spaces and pointed objects in a topos. Overall, the paper advances a 2-categorical and monadic framework to recognize and construct rectangular torsion theories and their bilimits, offering practical criteria and canonical decompositions.

Abstract

In this paper we introduce and study \emph{rectangular torsion theories}, i.e.\ those torsion theories with a pointed category, where the canonical functor is an equivalence of categories. In particular, we show that these are precisely the internal rectangular bands in the 2-category of pointed categories.

Paper Structure

This paper contains 4 sections, 33 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Rectangular pretorsion theories
  3. The pointed case
  4. A case study: torsion theory classes of epimorphisms

Key Result

Lemma 2.1

The diagram ( ) in $\mathbb{C}\times\mathbb{D}$, where $0\in\mathbf{0}$ and $1\in\mathbf{1}$, is a short exact sequence if and only if $m_1$ is an isomorphism, $m_2$ is a monomorphism, $e_1$ is an epimorphism and $e_2$ is an isomorphism.

Theorems & Definitions (71)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Example 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Lemma 2.7
  • ...and 61 more