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I gave the stub-entry Hopf algebroid a paragraph in the Idea-section that points out that already in commutative geometry there are two different kinds of Hopf algebroids associated with a groupoid (just as there are two versions of Hopf algebras associated with a group):
The commutative but non-co-commutative structure obtained by forming ordinary function algebras on objects and morphisms;
The non-commutative but co-commutative structure obtained by forming the groupoid convolution algebra.
For the moment I left the rest of the entry (which vaguely mentions commutative and non-commutative versions without putting them in relation) untouched, but I labelled the whole entry “under constructions”, since I think this issue needs to be discussed more for the entry not to be misleading.
I may find time to get back to this later…
It’s the same “oid” in both cases, as in: group with many objects.
But the algebraic notion generalizes away from smooth geometry to noncommutative geometry. Is that what you mean?
There is a brief hint of that in the entry. But the whole entry is a sad stub for the time being. Somebody should invest some energy into it.
Unlike in Hopf algebra theory, where the commutative/noncommutative aspect is about the total algebra, in Hopf algebroid community, the jargon is that noncommutative without further saying means noncommutative base, what is a difficult part in the theory; the theory over commutative base is a straightforward generalization of usual Hopf algebra theory. Part of the story is under bialgebroid. I think we should split the entry on the details of the case of commutative base (partly because the general case is so involved that it will itself fill the entry soon), I will do it.
added to Hopf algebroid in the examples a quick pointer to generalized Steenrod algebra (where I added some brief further details)
Added in the Examples-section on Generalized dual Steenrod algebra the precise statement (copied the same into Steenrod algebra):
+– {: .num_lemma #SelfHomologyIsModuleOverCohomologyRing}
Let $R$ be an E-∞ ring and let $A$ an E-∞ algebra over $R$. The self-generalized homology $A^R_\bullet(A)$ is naturally a module over the cohomology ring $A_\bullet$ via applying the homotopy groups $\infty$-functor $\pi_\bullet$ to the canonical inclusion
$A \stackrel{\simeq}{\rightarrow} A \underset{R}{\wedge} R \stackrel{}{\rightarrow} A \underset{R}{\wedge} A \,.$=–
+– {: .num_prop}
Let $R$ be an E-∞ ring and let $A$ an E-∞ algebra over $R$. If the the $A_\bullet$-module $A^R_\bullet(A)$ of lemma \ref{SelfHomologyIsModuleOverCohomologyRing} is a flat module, then
$(A_\bullet, A_\bullet(A))$ is a Hopf algebroid over $R_\bulllet$;
$A^R_\bullet(X)$ is a left $A^R_\bullet(A)$-module for every $R$-∞-module $X$.
=–
This is due to (Baker-Lazarev 01), further discussed in (Baker-Jeanneret 02) (there expressed in terms of the presentation by commutative monoids in symmetric spectra). A review is also in (Ravenel, chapter 2, prop. 2.2.8).
As I announced in 4. I have now split entry Hopf algebroid into Hopf algebroid over a commutative base (redirecting its more special case commutative Hopf algebra) and Hopf algebroid. As the technical discussion and categorical implication of the general case will require very long treatment (and the generalization is quite nontrivial even to define, and very interesting from the point of view of category and bicategory theory; of course large chunk of it is already at bialgebroid) the classical case over commutative base has a special entry; the higher versions from stable homotopy theory are commutative, graded commutative or higher commutative; I consequently changed the links in stable homotopy entries to refer to the entry on the classical case of commutative base. I also added a prefix commutative in theorems which hold only for commutative case. The higher geometry floating table of contents is also moved to commutative case, while higher algebra is retained in both entries.
Added a reference to Hopf algebroid over a commutative base.
A Tannaka duality-type theorem relating certain subcategory of commutative Hopf algebroids to discrete groupoids is in
- Laiachi EL Kaoutit, Representative functions on discrete groupoids and duality with Hopf algebroids, arxiv/1311.3109
I was chatting with Catherine Meusburger and Derek Wise about a project they’re doing with 3d TQFTs, when I suddenly realized that Hopf algebroids might be helpful. So, I needed to learn a bit about Hopf algebroids. It turns out that the Day-Street definition in Monoidal bicategories and Hopf algebroids is not equivalent to Gabriella Böhm’s definition, as had been asserted in the article here. I tried to fix that up and offer more details in the References section. Some of this material could be moved to the Definition section when somebody feels like continuing to write that.
Zoran, please check what you had in mind when writing “the two concepts are equivalent” in revision 3.
From Gabriella Böhm, Internal bialgebroids, entwining structures and corings, AMS Contemp. Math. 376 (2005) 207-226, math.QA/0311244
Bialgebroids have been characterized by B. Day and R. Street in purely categorical terms [DS]. They consider two objects $R$ and $A$ in a monoidal bicategory, a pseudo-monoid structure on $A$ together with a certain strong monoidal morphism from this pseudo-monoid to the canonical pseudo-monoid associated to $R$. In the particular case of the monoidal bicategory of bimodules – i.e. the one with 0-cells the $k$-algebras, 1-cells the bimodules and 2-cells the bimodule morphisms, – this definition recovers the one of the bialgebroid. As a support of our Definition \ref{bgd} of internal bialgebroids we prove an equivalence with the description of [DS] in the case of the monoidal bicategory of internal bimodules.
From G. Böhm, Hopf Algebroids, in: Handbook of algebra, arXiv/0805.3806v2:
A $^*$-autonomous structure on a strong monoidal special opmorphism between pseudomonoids in a monoidal bicategory
In the paper [DS], strong monoidal special opmorphisms $h$ in monoidal bicategories, from a canonical pseudomonoid $R^{op}\otimes R$ to some pseudomonoid $B$, were studied. The opmorphism $h$ was called Hopf if in addition there is a $^*$-autonomous structure on $B$ and $h$ is strong $^*$-autonomous (where $R^{op}\otimes R$ is meant to be $^*$-autonomous in a canonical way). In [DS], Section 3, a bialgebroid was described as a strong monoidal special opmorphism $h$ of pseudomonoids in the monoidal bicategory of [Algebras; Bimodules; Bimodule maps]. This opmorphism $h$ is strong $^*$-autonomous if and only if the corresponding bialgebroid constitutes a Hopf algebroid with a bijective antipode, see Section 4.2 of [hgdax].
[hgdax] Gabriella Böhm, Kornél Szlachányi, Hopf algebroids with bijective antipodes: axioms integrals and duals, J. Algebra 274 (2004), 708-750, math.QA/0302325
[DS] B. Day, R. Street, Quantum category, star autonomy and quantum groupoids, Fields Institute Comm. 43 pp 187-225, AMS, Providence, RI, 2004.
Finally, [hgdax]
We will show in Subsection \ref{DS} that the categorical definition [DS] of the Hopf bialgebroid is equivalent to our purely algebraic Definition \ref{def} of Hopf algebroid, apart from the tiny difference that we allow $S^2$ to be nontrivial on the base ring.
Theorem 4.7 says:
A strong $*$-autonomous structure on the bialgebroid $A$ over $L$ in the sense of [DS] is equivalent to a Hopf algebroid structure $(\mathcal{A}_L,S)$ in the sense of Definition \ref{def}.
I do not know what is the difference between [DS] and another reference of Day and Street (I looked more into another reference but this was years ago, when I did not understand the approach of Gabi).
Thanks, Zoran!
It would be great if all this were in the entry. As a first approximation, could you just paste this into a section “Properties – Relation between the definitions”?
I do hope John sees your reply here. If not, now that he edited the entry, we should contact him and hopefully somebody finds the time and energy to sort this out and give a decent discussion in the entry.
Yes, Urs, once we find the time and figure out the difference between the two different versions of Day and Street; maybe John’s disagreement is from that difference, and we actually all agree. I have two deadlines today, so not today.
I have contacted John, here is what he replies:
Ross Street said to me that his definition with Brian Day in the paper Zoran mentioned - “Monoidal bicategories and Hopf algebroids” - is not equivalent to Gabi Böhm’s later definition. However, he says that a later paper he wrote with Day contains a definition equivalent to Gabi Böhm’s definition. In fact this later paper appeared on the arXiv just a few days after hers - independently.
I’ve tried to edit the nLab page so that this is reasonably clear. Near the bottom I sketch Day and Street’s two definitions, which are very different.
Added axioms for the antipode in the simple case when it is invertible, due Böhm and Szlachányi (for noninvertible case, one needs a rather complicated axiomatics using a left and right bialgebroids). On the other hand, this definition avoid the ad hoc section map of Lu as an additional datum.
Yet another version of axiomatics for Hopf algebroids is in my preprint with Martina Stojić. A very obsolete version is at the arXiv:1610.03837, we are now trying to finish the revision to finally submit it somewhere, still several days of work. I shall of course replace the arXiv version when finished with the draft before sending to a journal. The title is now “Hopf algebroids with balancing subalgebra”.
The current version is twosha4.pdf (24 pages as of today, 18/06/20).
A definition of a variant of Hopf algebroid (over a noncommutative base) which is somewhat similar to Lu’s definition, but involves working with a 2-sided ideal, with help of a distinguished „balancing” subalgebra, is in
(just submitted, thoroughly extended v2 of April 12, 2021)
18: indeed, the revision took long time…
Further examples and counterexamples
The aim of this note is to communicate a simple example of a Lie–Rinehart algebra whose enveloping algebra is not a Hopf algebroid, neither in the sense of Böhm and Szlachányi, nor in the sense of Lu.
The concept of internal bialgebroids over a noncommutative base has been refined to an internal version of symmetric Hopf algebroids over a noncommutative base in
(today’s arXiv post)
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