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Hopf algebroids have a base and a total algebra (and some other data). There are 3 levels of generality: commutative Hopf algebras (classical case, both base and the total space are commutative), Hopf algebras with commutative base (studied from late 1980s: Maltsionitis, later Connes’ school etc) and genuinely noncommutative case for which see Hopf algebroid and bialgebroid.
Just as for groups and their Hopf algebras, there are two ways to assign a Hopf algebroid over a commutative base algebra to a groupoid $\mathcal{G}_\bullet$:
(commutative Hopf algebroids) with commutative but non-co-commutative total algebra: Form the commutative algebras of functions $C(\mathcal{G}_1)$ and $C(\mathcal{G}_0)$ and regard the operation induced by the partially defined composition in $\mathcal{G}_\bullet$ as an in general non-co-commutative coalgebra structure on $C(\mathcal{G}_1)$ over $C(\mathcal{G}_0)$; the graded commutative case appears in algebraic topology and is classical (Steenrod algebra and other examples);
with non-commutative but co-commutative total algebra: Form the in general non-commutative groupoid convolution algebra $C_{conv}(\mathcal{G})$ and regard it as a co-commutative coalgebra over $C(\mathcal{G}_0)$.
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).
under construction
We discuss here a natural generalization of the notion of groupoid convolution algebras to higher algebras for higher groupoids.
There may be several sensible such generalizations. The one discussed now follows the principle of iterated internalization and naturally matches to the concept of n-modules (n-vector spaces) as they appear in extended prequantum field theory.
In order to disentangle conceptual from technical aspects, we first discuss geometrically discrete higher groupoids. The results of this discussion then in particular help to suggest what the right definition of “higher Lie groupoid” in the ontext of higher convolution algebras should be in the first place.
The consideration are based on the following
By the discussion at 2-module we may think of the 2-category $k Alg_b$ of $k$-associative algebras and bimodules between them as a model for the 2-category 2Mod of $k$-2-modules that admit a 2-basis (2-vector spaces). Hence the groupoid convolution algebra constructiuon is a 2-functor
There is then the following systematic refinement of this to higher groupoids and higher algebra: by the discussion at n-module, 3-modules are algebra objects in 2Mod and maps between them are bimodule objects in there. An algebra object in $k Alg_b$ is equivalently a sesquialgebra, an algebra equipped with a second algebra structure up to coherent homotopy, that is exhibited by structure bimodules.
Special cases of this are bialgebras, for which these structure bimodules come from actual algebra homomorphisms. Examples of these in turn are Hopf algebras. These we naturally re-discover as special higher groupoid convolution higher algebras in example below.
This iterated internalization on the codomain of the groupoid convolution algebra functor has a natural analog on its domain: a 2-groupoid we may present by a double groupoid, namely a groupoid object in an (∞,1)-category in Grpd which is 3-coskeletal as a simplicial object in Grpd.
Given a groupoid object $\mathcal{G}_\bullet$ in the (2,1)-topos Grpd hence a double groupoid, applying the groupoid convolution algebra $(2,1)$-functor $C$ to the corresponding simplicial object $\mathcal{G}_\bullet \in Grpd^{\Delta^{op}}$ yields:
groupoid convolution algebras $C(\mathcal{G}_0)$ and $C(\mathcal{G}_1)$,
a $C(\mathcal{G}_1) \otimes_{C(\mathcal{G}_{0,1})} C(\mathcal{G}_1)-C(\mathcal{G}_{0})$-bimodule, assigned to the composition functor $\partial_1 \colon \mathcal{G}_1 \underset{\mathcal{G}_0}{\times} \mathcal{G}_1 \to \mathcal{G}_1$.
Under the 2-functoriality of $C$, the Segal conditions satisfied by $\mathcal{G}_\bullet$ make this bimodule exhibi a sesquialgebra structure over $C(\mathcal{G}_{0,1})$.
This sesquialgebra we call the the double groupoid convolution 2-algebra of $\mathcal{G}_\bullet$.
(Here we make invariant sense of the tensor product by evaluating on a Reedy fibrant representative.)
Let $G$ be a finite group. Write $\mathbf{B}G$ for its delooping groupoid (the connected groupoid with $\pi_1 = G$). There are two natural ways to present $\mathbf{B}G$ as a double groupoid:
$\underset{\longrightarrow}{\lim}(\cdots \mathbf{B}G \stackrel{\overset{id}{\to}}{\underset{id}{\to}} \mathbf{B}G) \simeq \mathbf{B}G$;
$\underset{\longrightarrow}{\lim}(\cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} *) \simeq \mathbf{B}G$.
Applying the groupoid convolution algebra functor to the first presentation yields the groupoid convolution algebra $C(\mathbf{B}G)$ equipped with a trivial multiplication bimodule, hence just the group convolution algebra $C(\mathbf{B}G) \simeq C_{conv}(G)$.
Applying however the groupoid convolution algebra functor to the second presentation yields the commutative algebra of functions $C(G)$ equipped with the multiplication bimodule which is $C(G \times G)$ regarded as a $(C(G\times G), C(G))$-bimdodule, where the right action is induced by pullback along the group product map $G \times G \to G$.
This bimodule is in the image of the functor $Alg \to Alg_b$ that sends algebra homomorphisms to their induced bimodules, by sending $f \colon A \to B$ to $A$ regarded as an $(A,B)$-bimodule with the canonical left action on itself and the right action induced by $f$. Namely this bimdoule correspondonds to the map
given on $\phi \in C(G)$ and $g_1, g_2 \in G$ by
In summary this means that (for $G$ a finite group)
if we regard $\mathbf{B}G$ as presented as a double groupoid constant on $\mathbf{B}G$, then the corresponding groupoid convolution sesquialgebra (basis for a 3-module) is the convolution algebra of $G$;
if instead we regard $\mathbf{B}G$ as presented as the double groupoid which is degreewise constant as a groupoid, then the corresponding groupoid convolution sesquialgebra is the standard (“dual”) Hopf algebra structure on the commutative pointwise product algebra of functions on $G$.
For $\Gamma$ a suitable commutative Hopf algebroid and $N_1, N_2$ two $\Gamma$-comodules, then the coTor groups $CoTor_\Gamma(N_1, N_2)$ form a Steenrod algebra. See there for details and citations.
For MU this is the content of the Landweber-Novikov theorem.
For $E$ a suitable E-infinity ring spectrum, its homotopy groups $\pi_\bullet(E)$ and generalized homology $E_\bullet(E)$ form a Hopf algebroid of spectra, the dual $E$-Steenrod algebra. (These examples have also been called brave new Hopf algebroids.) See at Steenrod algebra – Hopf algebroid structure.
The general statement is this:
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
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 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).
The generalization of commutative Hopf algebroids where the base is kept tcommutative while having the total algebra noncommutative, and the image of source and target maps are required to commute mutually is due Maltsiniotis; he also generalized this to quasi-Hopf version:
Rendus Acad. Sci. Paris 314, pp. 249-252 (1992) ps
A Tannaka duality-type theorem relating certain subcategory of commutative Hopf algebroids to discrete groupoids is in
For the relation to groupoid convolution algebras see also at groupoid convolution algebra – References – Convolution Hopf algebroids.
Last revised on February 28, 2021 at 16:41:37. See the history of this page for a list of all contributions to it.