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

Author(s)

Iusenko, Kostiantyn ^[1] ; MacQuarrie, John William ^[2] ; Quirino, Samuel ^{[2, 1]}

Affiliation

^[1] Univ Sao Paulo, Inst Matemat & Estat, Sao Paulo, SP - Brazil

^[2] Univ Fed Minas Gerais, Belo Horizonte, MG - Brazil

• Document type: Journal article
• Source: BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY; v. 52, n. 3, p. 697-719, SEP 2021.
• Web of Science Citations: 0

Abstract

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. (AU)

• FAPESP's process: 18/23690-6 - Structures, representations, and applications of algebraic systems
• Grantee: Ivan Chestakov
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 14/09310-5 - Algebraic structures and their representations
• Grantee: Vyacheslav Futorny
• Support Opportunities: Research Projects - Thematic Grants