Internal Hopf algebroid

Abstract: We introduce a natural generalization of the definition of a symmetric Hopf algebroid, internal to any symmetric monoidal category with coequalizers that commute with the monoidal product. Motivation for this is the study of Heisenberg doubles of countably dimensional Hopf algebras $A$ as internal Hopf algebroids over a (noncommutative) base $A$ in the category $\mathrm{indproVect}$ of filtered cofiltered vector spaces introduced by the author. One example of such Heisenberg double is internal Hopf algebroid $U(\mathfrak{g}) \sharp U(\mathfrak{g})^*$ over universal enveloping algebra $U(\mathfrak{g})$ of a finite-dimesional Lie algebra $\mathfrak{g}$ that is a properly internalized version of a completed Hopf algebroid previously studied as a Lie algebra type noncommutative phase space.