# Is the category of racks semi-abelian?

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

Any comments and references would be appreciated.

• It has no zero object (the initial object is empty while the terminal object is a singleton). So it's not semi-abelian.
– YCor
Jun 20, 2019 at 10:48
• I forgot to mention that the racks are pointed. So I edited the question. Thanks for your quick response. Jun 20, 2019 at 10:58
• Also there is a big difference between product and coproduct Jun 20, 2019 at 18:57
• Coproduct (also product) exists. And zero object as well. The missing explanations are whether being protomodular and Barr-exact. Jun 20, 2019 at 20:51
• @ToddTrimble: I presume Marco was thinking of the other established meaning of 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. Jun 21, 2019 at 13:24

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.