ON HOPFIAN AND CO-HOPFIAN OBJECTS IN AN ABELIAN CATEGORY AND IN THE CATEGORY OF ITS COMPLEX SEQUENCES OF MORPHISMS
Abstract
We study Hopfian and co-Hopfian objects in an abelian category and in the category of complex sequences of morphisms. Among other results, it is shown that if X and Y are two objects of a category 𝒜 and Hom𝒜(X, Y) is a co-Hopfian object such that every morphism in Hom𝒜(X, Y) is a monomorphism, then X is Hopfian; and if every morphism in Hom𝒜(X, Y) is an epimorphism, then Y is co-Hopfian. Furthermore, if 𝒜 is a balanced abelian category, X is a projective object of 𝒜 satisfying for every A ∈ O b(𝒜), A ≠ 0𝒜 implies Hom𝒜(X, A) ≠{eX, A}, and (Y, β) ∈ Ob(Comp(𝒜)) is such that HomComp(𝒜)(X,(Y, β)) is Hopfian, then (Y, β) is Hopfian. And, if 𝒜 is a balanced abelian category, Y is an injective object of 𝒜, and (X, α) ∈ O b(Comp(𝒜)) is such that HomComp(𝒜)((X, α), Y) is co-Hopfian, then (X, α) is co-Hopfian.
Related
- Pushforwards of lc pairs under morphisms to abelian varieties
- Examples of nontrivial presilting objects in a singularity category
- Kuranishi spaces in a category of fibrant objects and their homotopy fiber products
- Bounded t-structures on the category of strongly bounded objects
- Universal δ-functors and relative injective objects in the category of p-Banach spaces
BibTeX
@article{diallo2026hopfian,
title = {{ON HOPFIAN AND CO-HOPFIAN OBJECTS IN AN ABELIAN CATEGORY AND IN THE CATEGORY OF ITS COMPLEX SEQUENCES OF MORPHISMS}},
author = {Ablaye Diallo and Mohamed Ben Maaouia and Mamadou Sanghare},
journal = {JP Journal of Algebra, Number Theory and Applications},
year = {2026},
month = jul,
doi = {10.17654/0972555526041},
url = {https://hal.science/hal-05840267v1},
note = {CC BY},
}
RIS
TY - JOUR
TI - ON HOPFIAN AND CO-HOPFIAN OBJECTS IN AN ABELIAN CATEGORY AND IN THE CATEGORY OF ITS COMPLEX SEQUENCES OF MORPHISMS
AU - Ablaye Diallo
AU - Mohamed Ben Maaouia
AU - Mamadou Sanghare
PY - 2026
DA - 2026/07/25
JO - JP Journal of Algebra, Number Theory and Applications
DO - 10.17654/0972555526041
UR - https://hal.science/hal-05840267v1
AB - We study Hopfian and co-Hopfian objects in an abelian category and in the category of complex sequences of morphisms. Among other results, it is shown that if X and Y are two objects of a category 𝒜 and Hom_𝒜(X, Y) is a co-Hopfian object such that every morphism in Hom_𝒜(X, Y) is a monomorphism, then X is Hopfian; and if every morphism in Hom_𝒜(X, Y) is an epimorphism, then Y is co-Hopfian. Furthermore, if 𝒜 is a balanced abelian category, X is a projective object of 𝒜 satisfying for every A ∈ O b(𝒜), A ≠ 0_𝒜 implies Hom_𝒜(X, A) ≠{e_X, A}, and (Y, β) ∈ Ob(Comp(𝒜)) is such that Hom_Comp(𝒜)(X,(Y, β)) is Hopfian, then (Y, β) is Hopfian. And, if 𝒜 is a balanced abelian category, Y is an injective object of 𝒜, and (X, α) ∈ O b(Comp(𝒜)) is such that Hom_Comp(𝒜)((X, α), Y) is co-Hopfian, then (X, α) is co-Hopfian.
SN - 0972-5555
ER -