2021/07/23 by M. Ortiz-Morales, Ortiz-Morales, M., Rafael Ochoa +1
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2107.10982
openalex publication_date 2021/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we prove that the lower triangular matrix category Λ= [ \beginsmallmatrix T&0 M&U \endsmallmatrix ], where T and U are quasi-hereditary Hom-finite Krull-Schmidt K-categories and M is a \mathcal U⊗K \mathcal Top-module that satisfies suitable conditions, is quasi-hereditary in the sense of \citeLGOS1 and \citeMartin. Moreover, we solve the problem of finding quotients of path categories isomorphic to the lower triangular matrix category Λ, where \mathcal T=KR/J and \mathcal U=KQ/I are path categories of infinity quivers modulo admissible ideals. Finally, we study the case where Λ is a path category of a quiver Q with relations and \mathcal U is the full additive subcategory of Λ obtained by deleting a source vertex * in Q and \mathcal T=add \*\. We then show that there exists an adjoint pair of functors (\mathcal R, \mathcal E) between the functor categories mod Λ and mod \mathcal U that preserve orthogonality and exceptionality; see \citeAssem1. We then give some examples of how to extend classical tilting subcategories of \mathcal U-modules to classical tilting subcategories of Λ-modules.