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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 satisfying the identity $a\triangleleft b =a$, so it is a full epireflective subcategory of $\mathbf{Rack}$. In particular, there is an inclusion functor $\mathbf{Set}\... 2 First of all, let us see what is an algebra for this monad. One can show that the kernel of$(0,1):X\amalg G\to G$is generated by the conjugates of elements$X$by elements of$G$; so an action$\xi$is in some sense a way to interpret conjugation by$G$in$X$. To be more precise, we can define for all$g\in G$and$x\in Xg\ast x=\xi (gxg^{-1})\$; then ...