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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 ...

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 $\xi$ ...