Isoclinism and Stem Structures of 2-Groups

This study investigates the concept of isoclinism in the category of 2-groups by extending classical group-theoretic notions to higher categorical structures. Building on the categorical equivalence between crossed modules and 2-groups, the paper characterizes isoclinism for 2-groups through commutator maps and explores its key properties. Notably, it demonstrates that isoclinism forms an equivalence relation in the category of 2-groups, similar to the group and crossed module contexts. The paper further proves that every 2-group is isoclinic to a stem 2-group and establishes that isoclinism between 2-groups implies the corresponding isoclinism between their associated crossed modules. These results contribute to the broader understanding of homotopy-theoretic and categorical classifications within algebraic topology and category theory.