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Functorial embeddings associated with the Four Subspace Problem

Ivon Dorado, Gonzalo Medina

Abstract

We define a unified categorical framework for studying six subproblems arising from the classical Four Subspace Problem. For each subproblem, we construct a functor from its associated category to the category of representations of the quiver corresponding to the Four Subspace Problem. This approach gives a common structural setting for the six cases considered and allows a simultaneous and coherent analysis via functorial methods. We prove that the six functors are additive and fully faithful, and we show that none of them is dense. As a consequence, each functor induces an equivalence between the corresponding source category and a well-identified full subcategory of the target category. These equivalences provide an effective mechanism for transferring classification results and structural properties, thereby clarifying the structural interrelations among the categories studied.

Functorial embeddings associated with the Four Subspace Problem

Abstract

We define a unified categorical framework for studying six subproblems arising from the classical Four Subspace Problem. For each subproblem, we construct a functor from its associated category to the category of representations of the quiver corresponding to the Four Subspace Problem. This approach gives a common structural setting for the six cases considered and allows a simultaneous and coherent analysis via functorial methods. We prove that the six functors are additive and fully faithful, and we show that none of them is dense. As a consequence, each functor induces an equivalence between the corresponding source category and a well-identified full subcategory of the target category. These equivalences provide an effective mechanism for transferring classification results and structural properties, thereby clarifying the structural interrelations among the categories studied.

Paper Structure

This paper contains 7 sections, 19 theorems, 78 equations, 13 figures, 2 tables.

Table of Contents

  1. The Four Subspace Problem and its subproblems
  2. The quivers $\mathcal{S}$, $\mathcal{D}$, $\mathcal{K}$, and $\mathcal{C}$
  3. Categories associated to linear relations
  4. Six functors and their properties
  5. Definition of the functors
  6. Basic properties of the functors
  7. Matrix presentations

Key Result

Theorem 1

Given an arbitrary field $k$, all indecomposable representations of the quiver $\mathcal{F}$ are determined, up to isomorphism, by the objects presented in Figure fig:FSPsolution, together with the indecomposable injective objects listed in equ:injective.□

Figures (13)

  • Figure 1: The tetrad $\mathcal{T}$ and its one-point extension $\mathcal{F}$.
  • Figure 2: The quiver $\mathcal{S}$
  • Figure 3: The quiver $\mathcal{D}$
  • Figure 4: The quiver $\mathcal{K}$
  • Figure 5: The quiver $\mathcal{C}$
  • ...and 8 more figures

Theorems & Definitions (44)

  • Theorem 1
  • Theorem 2
  • proof
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Example
  • Lemma 7
  • proof
  • ...and 34 more