Homology, Homotopy and Applications

Semidirect products of categorical groups. Obstruction theory

Antonio R. Garzón and Hvedri Inassaridze

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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.

Article information

Source
Homology Homotopy Appl., Volume 3, Number 1 (2001), 111-138.

Dates
First available in Project Euclid: 19 February 2006

Permanent link to this document
https://projecteuclid.org/euclid.hha/1140370268

Mathematical Reviews number (MathSciNet)
MR1854641

Zentralblatt MATH identifier
0984.18005

Subjects
Primary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23]
Secondary: 18G50: Nonabelian homological algebra

Citation

Garzón, Antonio R.; Inassaridze, Hvedri. Semidirect products of categorical groups. Obstruction theory. Homology Homotopy Appl. 3 (2001), no. 1, 111--138. https://projecteuclid.org/euclid.hha/1140370268


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