A Functorial Approach to Gabrielk-quiver Constructions for Coalgebras and Pseudocompact Algebras

We define the path coalgebra and Gabriel quiver constructions as functors between the category ofk-quivers and the category of pointedk-coalgebras, forka field. We define a congruence relation on the coalgebra side, show that the functors above respect this relation, and prove that the induced Gabrielk-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 pseudocompact algebras in terms of path objects are unique, giving an application to homogeneous algebras.