Coherent and ideal actions in ideally exact categories

2025/07/08 by Manuel Mancini, Mancini, Manuel, Giuseppe Metere +2
Mathematics · #03B52 #06D35 #08C05 #16B50 #18C05 #18E13 #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2507.06124

openalex publication_date 2025/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

In the context of ideally exact categories, we introduce the notions of internal coherent action and internal ideal action that generalise different aspects of unital actions of rings and algebras. We prove that every ideal action is coherent, and that the converse statement holds in some relevant ideally exact contexts. Furthermore, a connection with G. Janelidze's notion of semidirect product in ideally exact categories is analysed.

Citations

• On actions and split extensions in varieties of hoops: the case of strong section
• Action accessible and weakly action representable varieties of algebras
• Ideally exact categories
• On the representability of actions of Leibniz algebras and Poisson algebras
• Nonmatrix varieties of nonassociative algebras
• Relative ideals in homological categories, with an application to MV-algebras

Related

• On the representability of actions of unital algebras
• Ideally regular categories
• Ideally exact categories
• Action representability of the category of internal groupoids
• Non-unital C^*-categories, (co)limits, crossed products and exactness