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# Homology, Homotopy and Applications

## Volume 3 (2001)

### Number 1

### Semidirect products of categorical groups. Obstruction theory

Pages: 111 – 138

DOI: http://dx.doi.org/10.4310/HHA.2001.v3.n1.a6

#### Authors

#### 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.

#### 2010 Mathematics Subject Classification

Primary 18D10. Secondary 18G50.