Triangular Matrix Categories over path Categories and Quasi-hereditary Categories, as well as one point extensions by Projectives

TL;DR

This paper investigates the structure of lower triangular matrix categories built from quasi-hereditary categories and path categories, establishing their properties, quotients, and relationships with subcategories and tilting theory.

Contribution

It proves that certain triangular matrix categories are quasi-hereditary, characterizes quotients of path categories, and constructs functor pairs that extend tilting subcategories.

Findings

Triangular matrix categories are quasi-hereditary under specified conditions.

Quotients of path categories can be isomorphic to these triangular matrix categories.

Existence of adjoint functor pairs that preserve orthogonality and extend tilting subcategories.

Abstract

In this paper, we prove that the lower triangular matrix category $\Lambda =\left [ \begin{smallmatrix} \mathcal{T}&0\\ M&\mathcal{U} \end{smallmatrix} \right ]$, where $\mathcal{T}$ and $\mathcal{U}$ are quasi-hereditary $\mathrm{Hom}$-finite Krull-Schmidt $K$-categories and $M$ is a $\mathcal U\otimes_K \mathcal T^{op}$-module that satisfies suitable conditions, is quasi-hereditary in the sense of \cite{LGOS1} and \cite{Martin}. Moreover, we solve the problem of finding quotients of path categories isomorphic to the lower triangular matrix category $\Lambda$, where $\mathcal T=K\mathcal{R/J}$ and $\mathcal U=K\mathcal{Q/I}$ are path categories of infinity quivers modulo admissible ideals. Finally, we study the case where $\Lambda$ is a path category of a quiver $Q$ with relations and $\mathcal U$ is the full additive subcategory of $\Lambda$ obtained by deleting a source vertex $*$ in $Q$ and $\mathcal T=\mathrm{add} \{*\}$. We then show that there exists an adjoint pair of functors $(\mathcal R, \mathcal E)$ between the functor categories $\mathrm{mod} \ \Lambda$ and $\mathrm{mod} \ \mathcal U$ that preserve orthogonality and exceptionality; see \cite{Assem1}. We then give some examples of how to extend classical tilting subcategories of $\mathcal U$-modules to classical tilting subcategories of $\Lambda$-modules.