Abstract
In this work, we construct a relationship between matched pairs and triples of groupoids. Given two 3-groupoids with a common edge, we construct a triple groupoid by using the matched pairs actions.
Highlights
Matched pairs of groups were introduced by Takeuchi [17] as a group version of Singer’s work [16] for Hopf algebras
The theory of matched pairs was used as a tool for set theoretic solutions of the Yang–Baxter equation in [10]
In this work, following Brown, we investigate this situation for triple groupoids, diagrammatically
Summary
Matched pairs of groups were introduced by Takeuchi [17] as a group version of Singer’s work [16] for Hopf algebras. Definition 2.1 A matched pair of groups means a triple (G1, G2, σ) where G1 and G2 are groups and the map σ : G1 × G2 → G2 × G1 (g1, g2) → (g1 ⇀ g2, g1 ↼ g2) If G1 and G2 are subgroups of a group G such that the product map G1 × G2 → G is bijective, (G1, G2) forms a matched pair with structure σ (g1, g2) = (g1 ⇀ g2, g1 ↼ g2) defined by g1g2 = (g1 ⇀ g2) (g1 ↼ g2) .
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