I wonder whether the category of (pointed) racks is semi-abelian.
Any comments and references would be appreciated.
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3$\begingroup$ It has no zero object (the initial object is empty while the terminal object is a singleton). So it's not semi-abelian. $\endgroup$– YCorCommented Jun 20, 2019 at 10:48
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2$\begingroup$ I forgot to mention that the racks are pointed. So I edited the question. Thanks for your quick response. $\endgroup$– Kadir EmirCommented Jun 20, 2019 at 10:58
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1$\begingroup$ Also there is a big difference between product and coproduct $\endgroup$– Marco FarinatiCommented Jun 20, 2019 at 18:57
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1$\begingroup$ Coproduct (also product) exists. And zero object as well. The missing explanations are whether being protomodular and Barr-exact. $\endgroup$– Kadir EmirCommented Jun 20, 2019 at 20:51
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1$\begingroup$ @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. $\endgroup$– Peter LeFanu LumsdaineCommented Jun 21, 2019 at 13:24
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1 Answer
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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.
