Path coalgebra as a right adjoint functor

Abstract

We define a certain variant of the category of quivers and construct thepath coalgebra as a functor, the main tool for this process being the universal property of cotensor coalgebras. Then we construct a functor from the category of pointed coalgebras to this category of quivers, based on the Gabriel quiver of pointed coalgebras. With a relation on the morphisms of the category of pointed coalgebras we obtain an adjunction between these two functors.