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 $Aff_k$ of affine algebraic $k$-schemes, where $k$ is a field, the $k$-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 due to Day and Street, and another due Gabi Böhm, using pairs of a left and right bialgebroid. Gabi later showed that the two definitions are in fact equivalent.
Given a Hopf algebra $B$ and a braided-commutative algebra $A$ in the category of Yetter-Drinfeld modules over $B$, the smash product algebra $B\sharp A$ is the total algebra of a Hopf algebroid over $A$. 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 has been discovered by several different people in several different formalisms. Some of the differences are merely cosmetic, but there are at least two main concepts, depending on the underlying concept of ‘bialgebroid’.
Day and Street have a concept of Hopf algebroid here:
In this they start by taking an algebroid to be an “algebra with several objects” in the sense of a $k$-linear category $A$: that is, a $V$-enriched category with $V = Vect_k$. The 2-category $V Cat$ of $k$-linear categories, functors and natural transformations is monoidal (where the tensor product of $V$-categories is defined by cartesian product on object sets and tensor product on hom-spaces). So, they define a bialgebroid to be a comonoid in $V Cat$. Because the tensor product is cartesian product on object sets, the comultiplication in such a bialgebroid is forced to be the diagonal on objects. Thus, their notion of bialgebroid amounts to a $k$-linear category $A$ equipped with linear maps
satisfying coassociativity, a version of the usual bialgebra axiom, and so on. On page 142 of the above reference they define an antipode on a bialgebroid $A$ to be a $k$-linear functor $S: A \to A^{op}$ together with a natural isomorphism
A Hopf algebroid is then roughly a bialgebroid with an antipode. With this definition, a Hopf algebra gives a one-object Hopf algebroid.
A different and more widely used concept was developed independently in these two papers, which appeared on the arXiv within a couple of days of each other:
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
R. Street and B. Day, Quantum categories, star autonomy, and quantum groupoids, in “Galois Theory, Hopf Algebras, and Semiabelian Categories”, Fields Institute Communications 43 (American Math. Soc. 2004) 187-226; arXiv:0301209
and also described in:
A class of examples of such Hopf algebroids internally in a monoidal category of cocomplete cofiltered vector spaces is in
This starts with a different concept of bialgebroid, which is discussed here on the nLab. Namely: any $k$-algebra $R$ gives a pseudomonoid $R^e = R^{op} \otimes R$ in the bicategory $Mod_k$ of k-algebras, bimodules, and bimodule homomorphisms, and a bialgebroid is then an opmonoidal monad $A$ on $R^e$. When the fusion (or Galois) operator for this opmonoidal monad is invertible, we say that $A$ is a Hopf algebroid. In G. Böhm’s work this definition is stated in a less compressed, more down-to-earth way.