Given an internal groupoid in the category of affine algebraic -schemes, where is a field, the -algebras of global sections over the scheme of objects and scheme of morphisms have an additional structure of a commutative Hopf algebroid. In fact this is an antiequivalence of categories. Commutative Hopf algebroids are useful also in a version in brave new algebra? (the work of Rognes).
There are several generalization to the noncommutative case. A difficult part is to work over the noncommutative base (i.e., the object of objects is noncommutative). The definition of a bialgebroid? is not that difficult and there is even a very old definition due Takeuchi. To add an antipode is nontrivial. A definition of Lu from mid 1990s is rather nonselfdual unlike the case of Hopf algebras. So a better solution is to abandon the idea of an antipode and have some replacement for it. There are two approaches, one via monoidal bicategories? due to Day and Street, and another due Gabi Böhm, using pairs of a left and right bialgebroid. Gabi has later shown that the two definitions are in fact equivalent.
B. Day, R. Street, Monoidal bicategories and Hopf algebroids, Advances in Mathematics, 129, 1 (1997) 99–157
G. Böhm, An alternative notion of Hopf algebroid; in “Hopf algebras in noncommutative geometry and physics”, 31–53, Lecture Notes in Pure and Appl. Math., 239, Dekker, New York, 2005; math.QA/0301169