arXiv subjects

Frank Murphy-Hernandez

Publications and source records attributed to Frank Murphy-Hernandez.

4 recordsLinked to original sources

arXiv2026

The Grouplike Functor and Monoidal Adjunctions

We construct a categorification of the functor of grouplike elements of a Hopf algebra in the setting of braided monoidal categories. Given a braided bistrong monoidal functor admitting a right adjoint, we define a functor assigning to each Hopf monoid a group object obtained as a subobject of its underlying object by means of finite limits. We prove that this construction is functorial and provides a right adjoint to the induced free Hopf monoid functor, thereby extending the classical correspondence between groups and grouplike elements of Hopf algebras to a categorical setting.

math.CT↗

arXiv2026

The Cofree Functor on G-Sets and Its Approximations

We study the cofree functor on the category of group actions and examine its categorical and combinatorial properties. Motivated by this construction, we introduce a family of functors associated with subgroups and develop their theory through a corresponding coreflective subcategory of group actions. Finally, we investigate the comonad in the category of sets induced by the cofree adjunction and prove that it classifies the groups.

math.CT↗

arXiv2025

Classes of Preradicals Induced by Relative Injectivity

This paper analyzes new classes of preradicals defined as weak forms of left exact and idempotent preradicals. We introduce prehereditary preradicals to generalize the hereditary property, and essentially idempotent and weakly idempotent preradicals to generalize idempotency. Our motivation is to study the concept of relative injectivity. We show that these new classes facilitate the extension of classical results on injectivity with respect to a hereditary torsion theory to a broader context, requiring weaker assumptions.

math.RA↗

arXiv2025

Matricial Closure

We study a closure operator derived from the matrix endofunctor on the category of rings with unity. We investigate the invariance of various ring-theoretic properties under this operator. A key finding is the decisive nature of this operator: a property is either preserved for all rings or fails for all rings. This work provides a comprehensive analysis of the behavior of ring properties under matricial closure, including chain conditions, radical-theoretic properties, and other structural characteristics.

math.RA↗