From Rota–Baxter coalgebras to (tri)dendriform coalgebras

This paper studies the relationship among Rota–Baxter coalgebras, dendriform coalgebras, and tridendriform coalgebras, whose corresponding algebraic structures were initiated by Aguiar. We provide some examples of these coalgebraic structures, investigate the duality with (tri)dendriform algebras via linear and finite duals, and present a construction of (tri)dendriform coalgebras from Rota–Baxter coalgebras of arbitrary weight. We end with a commutative diagram of functors that links the categories of Rota–Baxter coalgebras to those of (tri)dendriform coalgebras.