Semidirect products of categorical groups. Obstruction theory

Abstract

By considering the notion of action of a categorical group ${\mathbb G}$ on another categorical group ${\mathbb H}$ we define the semidirect product ${\mathbb H}\ltimes {\mathbb G}$ and classify the set of all split extensions of ${\mathbb G}$ by ${\mathbb H}$. Then, in an analogous way to the group case, we develop an obstruction theory that allows the classification of all split extensions of categorical groups inducing a given pair $(\varphi,\psi)$ (called a collective character of ${\mathbb G}$ in ${\mathbb H}$) where $\varphi:\pi_0({\mathbb G})\rightarrow \pi_0({\cal E}q({\mathbb H}))$ is a group homomorphism and $\psi:\pi_1({\mathbb G})\rightarrow \pi_1({\cal E}q({\mathbb H}))$ is a homomorphism of $\pi_0({\mathbb G})$-modules.