Automorphic objects in categories (Q5932609)

The author introduces and studies automorphic objects in categories. There are analogs of group objects with a group replaced by an automorphic set in the sense of \textit{E.~Brieskorn}. The latter is a set with a product for which left translations are automorphisms. In particular, the author proves the following theorem: In a category \({\mathcal C}\) with finite products, an object \(c\) is automorphic if and only if the \(\operatorname{hom}\) functor \({\mathcal C}(-,c)\) is an automorphic object in the functor category \({\mathcal S}et^{{\mathcal C}^{op}}\). In the last section, the author presents various examples of automorphic objects, connected with racks and quandles arising in low-dimensional topology.