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Semiabelian categories are pointed exact protomodular categories with coproducts. Suitable for questions about protomodular categories as well.
13
votes
Is the category of racks semi-abelian?
The category $\mathbf{Rack}$ of racks is Barr-exact since it is a variety of universal algebras, but it is not protomodular. Indeed, the category of sets is equivalent to the category of racks satisfy …
2
votes
Accepted
Relating three viewpoints on the semidirect product
First of all, let us see what is an algebra for this monad. One can show that the kernel of $(0,1):X\amalg G\to G$ is generated by the conjugates of elements $X$ by elements of $G$; so an action $\xi$ …