# The path algebra as a left adjoint functor

## TL;DR

This paper introduces a categorical framework connecting finite quivers and pseudocompact associative algebras, defining functors for path algebras and Gabriel quivers, and establishing an adjoint pair that respects ideals and induces an equivalence.

## Contribution

It constructs an intermediate category and explicit functors that form an adjoint pair, linking quivers and algebras in a novel categorical setting.

## Key findings

- Defined the completed path algebra and Gabriel quiver as functors.
- Established an adjoint pair of functors respecting ideals.
- Proved an equivalence between categories related to finite quivers and pseudocompact algebras.

## Abstract

We consider an intermediate category between the category of finite quivers and a certain category of pseudocompact associative algebras whose objects include all pointed finite dimensional algebras. We define the completed path algebra and the Gabriel quiver as functors. We give an explicit quotient of the category of algebras on which these functors form an adjoint pair. We show that these functors respect ideals, obtaining in this way an equivalence between related categories.