symmetric monoidal (∞,1)-category of spectra
A Hopf algebroid is an associative bialgebroid with an antipode.
A Hopf algebroid is a (possibly noncommutative) generalization of a structure which is dual to a groupoid (equipped with atlas) in the sense of space-algebra duality. This is the concept that generalizes Hopf algebras with their relation to groups from groups to groupoids.
More details on the case of the Hopf algebroids associated to groupoids, i.e. the convolution Hopf algebroid and groupoid Hopf algebroid, as well as stable homotopy examples like Steenrod algebra, see entry Hopf algebroid over a commutative base.
In the general case we should distinguish left and right bialgebroids, see bialgebroid.
In one of the versions, a general Hopf algebroid is defined as a pair of a left algebroid and right algebroid together with a linear map from left to right bialgebroid taking the role of an antipode (…).
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 the 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 John Rognes).
There are several generalizations 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.
Given a Hopf algebra and a braided-commutative algebra in the category of Yetter-Drinfeld modules over , the smash product algebra is the total algebra of a Hopf algebroid over . This is a noncommutative generalization (of formal dual of) an action groupoid.
The commutative version is classical, and there is an extensive literature, see Hopf algebroids over a commutative base.
Over a noncommutative base ring, there is a nonsymmetric version due J-H. Lu and a similar version is later studied by Ping Xu
The modern concept over the noncommutative base is discovered in two formally different formalisms, but the two concepts are equivalent: