I wonder whether the category of (pointed) racks is semi-abelian.

Any comments and references would be appreciated.

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I wonder whether the category of (pointed) racks is semi-abelian.

Any comments and references would be appreciated.

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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 $\mathbf{Set}\to \mathbf{Rack}$ which preserves limits and reflects isomorphisms; then if $\mathbf{Rack}$ was protomodular $\mathbf{Set}$ would also be protomodular, which is false.

I found this argument in the paper "A Galois-Theoretic Approach to the Covering Theory of Quandles" by Valérian Even. It also shows that $\mathbf{Rack}$ cannot be Mal'tsev, or even congruence-permutable.

semi-abelian, due (as far as I can see) to Palamodov, given on Wikipedia and discussed on MO here. Palamodov’s sense does require biproducts, so Marco’s comment shows racks aren’t semi-abelian in that sense either. But it’s clear from Kadir’s second comment that he has the more standard meaning in mind. $\endgroup$ – Peter LeFanu Lumsdaine Jun 21 '19 at 13:243more comments