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The normal decomposition of a morphism in categories without zeros

Renier Jansen, Muhammad Qasim, Walter Tholen

Abstract

For a morphism f in a category C with sufficiently many finite limits and colimits, we discuss an elementary construction of a decomposition of f through objects P and N which, if C happens to have a zero object, amounts to the standard decomposition of f through P = Coker(ker f) and N = Ker(coker f). In this way we obtain natural notions of normal monomorphism and normal epimorphism also in non-pointed categories, as special types of regular mono- and epimorphisms. We examine the factorization behaviour of these classes of morphisms in general, compare the generalized normal decompositions with other types of threefold factorizations, and illustrate them in some every-day categories. The concrete construction of normal decompositions in the slices or coslices of these categories can be challenging. Amongst many others, in this regard, we consider particularly the categories of T1-spaces and of groups.

The normal decomposition of a morphism in categories without zeros

Abstract

For a morphism f in a category C with sufficiently many finite limits and colimits, we discuss an elementary construction of a decomposition of f through objects P and N which, if C happens to have a zero object, amounts to the standard decomposition of f through P = Coker(ker f) and N = Ker(coker f). In this way we obtain natural notions of normal monomorphism and normal epimorphism also in non-pointed categories, as special types of regular mono- and epimorphisms. We examine the factorization behaviour of these classes of morphisms in general, compare the generalized normal decompositions with other types of threefold factorizations, and illustrate them in some every-day categories. The concrete construction of normal decompositions in the slices or coslices of these categories can be challenging. Amongst many others, in this regard, we consider particularly the categories of T1-spaces and of groups.

Paper Structure

This paper contains 9 sections, 32 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Reflective and coreflective classes of morphisms
  3. Normal monomorphisms and epimorphisms, the normal closure
  4. The normal decomposition of a morphism
  5. Orthogonal threefold factorization systems
  6. Normal decomposition in slice and coslice categories
  7. Slices and coslices of the category of T$_1$-spaces
  8. Slices and coslices of the category of groups
  9. Concluding remarks on the pullback-pushout adjunction

Key Result

Proposition 2.3

A replete class $\mathcal{M}$ of $\mathds{C}$ is reflective if, and only if, every morphism $f$ in $\mathds{C}$ factors as $f=n_f\cdot \hat{f}$ with $n_f\in\mathcal{M}$ such that, whenever $m\cdot u=v\cdot f$ with $m\in\mathcal{M}$, one has $t\cdot \hat{f}=u$ and $m\cdot t=v\cdot n_f$, for a uniquel

Theorems & Definitions (55)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • Definition 3.1
  • Theorem 3.2
  • ...and 45 more