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Galois theory and homology in quasi-abelian functor categories

Nadja Egner

Abstract

Given a finite category T, we consider the functor category [T,A], where A can in particular be any quasi-abelian category. Examples of quasi-abelian categories are given by any abelian category but also by non-exact additive categories as the categories of torsion(-free) abelian groups, topological abelian groups, locally compact abelian groups, Banach spaces and Fréchet spaces. In this situation, the categories of various internal categorical structures in A, such as the categories of internal n-fold groupoids, are equivalent to functor categories [T,A] for a suitable category T. For a replete full subcategory S of T, we define F to be the full subcategory of [T,A] whose objects are given by the functors G with G(X)=0 for all objects X not in S. We prove that F is a torsion-free Birkhoff subcategory of [T,A]. This allows us to study (higher) central extensions from categorical Galois theory in [T,A] with respect to F and generalized Hopf formulae for homology.

Galois theory and homology in quasi-abelian functor categories

Abstract

Given a finite category T, we consider the functor category [T,A], where A can in particular be any quasi-abelian category. Examples of quasi-abelian categories are given by any abelian category but also by non-exact additive categories as the categories of torsion(-free) abelian groups, topological abelian groups, locally compact abelian groups, Banach spaces and Fréchet spaces. In this situation, the categories of various internal categorical structures in A, such as the categories of internal n-fold groupoids, are equivalent to functor categories [T,A] for a suitable category T. For a replete full subcategory S of T, we define F to be the full subcategory of [T,A] whose objects are given by the functors G with G(X)=0 for all objects X not in S. We prove that F is a torsion-free Birkhoff subcategory of [T,A]. This allows us to study (higher) central extensions from categorical Galois theory in [T,A] with respect to F and generalized Hopf formulae for homology.
Paper Structure (10 sections, 9 theorems, 14 equations)

This paper contains 10 sections, 9 theorems, 14 equations.

Table of Contents

  1. Torsion theory $(\mathscr{T},\mathscr{F})$ in $\mathscr{A}^\mathbb{T}$
  2. Torsion theories
  3. Torsion theory $(\mathscr{T},\mathscr{F})$ in $\mathscr{A}^\mathbb{T}$
  4. Central extensions in $\mathscr{A}^\mathbb{T}$ with respect to $\mathscr{F}$
  5. Categorical Galois theory
  6. Categorical theory of central extensions
  7. Central extensions in $\mathscr{A}^\mathbb{T}$ with respect to $\mathscr{F}$
  8. Higher central extensions and generalized Hopf formulae in $\mathscr{A}^\mathbb{T}$ with respect to $\mathscr{F}$
  9. Higher central extensions and generalized Hopf formulae
  10. Higher central extensions and generalized Hopf formulae in $\mathscr{A}^\mathbb{T}$ with respect to $\mathscr{F}$

Key Result

Proposition 1.1

Let $\mathscr{C}$ be a pointed category with kernels and pullback stable normal epimorphisms, and $\mathscr{F}$ be a replete full subcategory of $\mathscr{C}$. Then the following conditions are equivalent:

Theorems & Definitions (24)

  • Proposition 1.1
  • Proposition 1.2
  • Remark 1.3: Quasi-abelian categories
  • Proposition 1.4
  • proof
  • Remark 1.5
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 14 more