Moduli problems are extensively studied in mathematics, for example moduli of semi/stable bundles on curves, semi/stable coherent sheaves on varieties, representations of algebras. These examples can all be understood as studying moduli of objects in abelian categories. A classical result of King is that the moduli space of semistable representations of acyclic quivers (and more generally, finite-dimensional algebras) is a projective variety. Deligne's theorem asserts that the category of finite-dimensional representations of a finite-dimensional algebra can be described purely categorically; in other words, any finite abelian category arises in this manner. We tried to remove one of the assumptions on finite categories and see if a moduli space of objects can be constructed in this generality using the recent advances of the theory of good moduli spaces. However, we found out that without the extra assumption, the moduli problem may not even form an algebraic stack.
This talk is based on an ongoing discussion with Andres Herrero, Andrew Kwon and Emma Lennen, and it will cover quiver representations as a motivating example, definition of finite categories and moduli problems in abelian categories, and a counterexample that we found.