  Recently, Li, Sheng and Tang introduced post-Hopf algebras and relative
Rota-Baxter operators (on cocommutative Hopf algebras), providing an adjunction
between the respective categories under the assumption that the structures
involved are cocommutative. We introduce Yetter-Drinfeld post-Hopf algebras,
which become usual post-Hopf algebras in the cocommutative setting. In analogy
with the correspondence between cocommutative post-Hopf algebras and
cocommutative Hopf braces, the category of Yetter-Drinfeld post-Hopf algebras
is isomorphic to the category of Yetter-Drinfeld braces introduced by the
author in a joint work with D. Ferri. This allows to explore the connection
with matched pairs of actions and provide examples of Yetter-Drinfeld post-Hopf
algebras. Moreover, we prove that the category of Yetter-Drinfeld post-Hopf
algebras is equivalent to a subcategory of Yetter-Drinfeld relative Rota-Baxter
operators. The latter structures coincide with the inverse maps of
Yetter-Drinfeld 1-cocycles introduced by the author and D. Ferri, and
generalise bijective relative Rota-Baxter operators on cocommutative Hopf
algebras. Hence the previous equivalence passes to cocommutative post-Hopf
algebras and bijective relative Rota-Baxter operators. Once the surjectivity of
the Yetter-Drinfeld relative Rota-Baxter operators is removed, the equivalence
is replaced by an adjunction and one can recover the result of Li, Sheng and
Tang in the cocommutative case.
