In this paper we introduce the notion of a-monoidal distributive law between two Hopf quasigroups A and H in a symmetric monoidal category. We prove that every a-monoidal distributive law induces a product on A ⊗ H, called the wreath product, such that A ⊗ H becomes a Hopf quasigroup. Finally, we show that several well-known constructions fit naturally into this framework. In particular, double cross products of Hopf quasigroups, cross products of Hopf quasigroups with a skew pairing between them, smash products of Hopf quasigroups and twisted smash products of Hopf quasigroups are examples of wreath products associated to a-monoidal distributive laws.
Keywords: Hopf quasigroup, loop, distributive law, comonoidal, wreath product
2020 MSC: 18M05, 20N05, 16T99
Theory and Applications of Categories, Vol. 45, 2026, No. 47, pp 2034-2063.
Published 2026-09-16.
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