2016/05/27 by Mathieu Duckerts-Antoine, Duckerts-Antoine, Mathieu, Marino Gran +3
Mathematics · #08C15 #18A20 #18A22 #18A32 #18A40 #18D30 #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1605.08627
openalex publication_date 2016/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
On a category \mathscrC with a designated (well-behaved) class M of monomorphisms, a closure operator in the sense of D. Dikranjan and E. Giuli is a pointed endofunctor of M, seen as a full subcategory of the arrow-category \mathscrC2 whose objects are morphisms from the class M, which "commutes" with the codomain functor cod\colon M→ \mathscrC. In other words, a closure operator consists of a functor C\colon M\toM and a natural transformation c\colon 1M→ C such that cod ⋅ C=C and cod⋅ c=1cod. In this paper we adapt this notion to the domain functor dom\colon E→\mathscrC, where E is a class of epimorphisms in \mathscrC, and show that such closure operators can be used to classify E-epireflective subcategories of \mathscrC, provided E is closed under composition and contains isomorphisms. Specializing to the case when E is the class of regular epimorphisms in a regular category, we obtain known characterizations of regular-epireflective subcategories of general and various special types of regular categories, appearing in the works of the second author and his coauthors. These results show the interest in investigating further the notion of a closure operator relative to a general functor. They also point out new links between epireflective subcategories arising in algebra, the theory of fibrations, and the theory of categorical closure operators.