# A functorial approach to Gabriel $k$-quiver constructions for coalgebras and pseudocompact algebras

## TL;DR

This paper develops a functorial framework linking $k$-quivers with pointed $k$-coalgebras and pseudocompact algebras, establishing adjoint pairs and analyzing presentation uniqueness.

## Contribution

It introduces functorial constructions for Gabriel $k$-quivers and path coalgebras, including dualizations for pseudocompact algebras, and explores their categorical properties.

## Key findings

- Gabriel $k$-quiver functor is left adjoint to path coalgebra functor
- Established dual adjoint pairs for pseudocompact algebras
- Characterized the uniqueness of algebra and coalgebra presentations

## Abstract

We define the path coalgebra and Gabriel quiver constructions as functors between the category of $k$-quivers and the category of pointed $k$-coalgebras, for $k$ a field. We define a congruence relation on the coalgebra side, show that the functors above respect this relation, and prove that the induced Gabriel $k$-quiver functor is left adjoint to the corresponding path coalgebra functor. We dualize, obtaining adjoint pairs of functors (contravariant and covariant) for pseudocompact algebras. Using these tools we describe precisely to what extent presentations of coalgebras and algebras in terms of path objects are unique, giving an application to homogeneous algebras.