Functorial embeddings associated with the Four Subspace Problem

TL;DR

This paper introduces a categorical framework for analyzing six subproblems of the Four Subspace Problem, establishing functorial relationships that clarify their structural interrelations and facilitate classification.

Contribution

It constructs six functors from subproblem categories to quiver representations, proving their additivity, full faithfulness, and providing categorical equivalences.

Findings

Functors are additive and fully faithful.

No functor is dense, indicating distinct subcategory embeddings.

Framework enables transfer of classification results.

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.