  A finite directed category is a $k$-linear category with finitely many
objects and an underlying poset structure, where $k$ is an algebraically closed
field. This concept unifies structures such as $k$-linerizations of posets and
finite EI categories, quotient algebras of finite-dimensional hereditary
algebras, triangular matrix algebras, etc. In this paper we study
representations of finite directed categories, discuss their stratification
properties, and show the existence of generalized APR tilting modules for
triangular matrix algebras under some assumptions.
