nLab Hopf algebroid over a commutative base



Higher algebra

Higher geometry


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 𝒢 \mathcal{G}_\bullet:

  1. (commutative Hopf algebroids) with commutative but non-co-commutative total algebra: Form the commutative algebras of functions C(𝒢 1)C(\mathcal{G}_1) and C(𝒢 0)C(\mathcal{G}_0) and regard the operation induced by the partially defined composition in 𝒢 \mathcal{G}_\bullet as an in general non-co-commutative coalgebra structure on C(𝒢 1)C(\mathcal{G}_1) over C(𝒢 0)C(\mathcal{G}_0); the graded commutative case appears in algebraic topology and is classical (Steenrod algebra and other examples);

  2. with non-commutative but co-commutative total algebra: Form the in general non-commutative groupoid convolution algebra C conv(𝒢)C_{conv}(\mathcal{G}) and regard it as a co-commutative coalgebra over C(𝒢 0)C(\mathcal{G}_0).

Given an internal groupoid in the category Aff kAff_k of affine algebraic kk-schemes, where kk is a field, the kk-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).

Higher groupoid convolution algebras and n-vector spaces/n-modules

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 kAlg bk Alg_b of kk-associative algebras and bimodules between them as a model for the 2-category 2Mod of kk-2-modules that admit a 2-basis (2-vector spaces). Hence the groupoid convolution algebra constructiuon is a 2-functor

C:Grpd2Mod. C \;\colon\; Grpd \to 2 Mod \,.

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 kAlg bk 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 𝒢 \mathcal{G}_\bullet in the (2,1)-topos Grpd hence a double groupoid, applying the groupoid convolution algebra (2,1)(2,1)-functor CC to the corresponding simplicial object 𝒢 Grpd Δ op\mathcal{G}_\bullet \in Grpd^{\Delta^{op}} yields:

  • groupoid convolution algebras C(𝒢 0)C(\mathcal{G}_0) and C(𝒢 1)C(\mathcal{G}_1),

  • a C(𝒢 1) C(𝒢 0,1)C(𝒢 1)C(𝒢 0)C(\mathcal{G}_1) \otimes_{C(\mathcal{G}_{0,1})} C(\mathcal{G}_1)-C(\mathcal{G}_{0})-bimodule, assigned to the composition functor 1:𝒢 1×𝒢 0𝒢 1𝒢 1\partial_1 \colon \mathcal{G}_1 \underset{\mathcal{G}_0}{\times} \mathcal{G}_1 \to \mathcal{G}_1.

Under the 2-functoriality of CC, the Segal conditions satisfied by 𝒢 \mathcal{G}_\bullet make this bimodule exhibi a sesquialgebra structure over C(𝒢 0,1)C(\mathcal{G}_{0,1}).

This sesquialgebra we call the the double groupoid convolution 2-algebra of 𝒢 \mathcal{G}_\bullet.

(Here we make invariant sense of the tensor product by evaluating on a Reedy fibrant representative.)


Let GG be a finite group. Write BG\mathbf{B}G for its delooping groupoid (the connected groupoid with π 1=G\pi_1 = G). There are two natural ways to present BG\mathbf{B}G as a double groupoid:

  1. lim(BGididBG)BG\underset{\longrightarrow}{\lim}(\cdots \mathbf{B}G \stackrel{\overset{id}{\to}}{\underset{id}{\to}} \mathbf{B}G) \simeq \mathbf{B}G;

  2. lim(G×GG*)BG\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(BG)C(\mathbf{B}G) equipped with a trivial multiplication bimodule, hence just the group convolution algebra C(BG)C conv(G)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)C(G) equipped with the multiplication bimodule which is C(G×G)C(G \times G) regarded as a (C(G×G),C(G))(C(G\times G), C(G))-bimdodule, where the right action is induced by pullback along the group product map G×GGG \times G \to G.

This bimodule is in the image of the functor AlgAlg bAlg \to Alg_b that sends algebra homomorphisms to their induced bimodules, by sending f:ABf \colon A \to B to AA regarded as an (A,B)(A,B)-bimodule with the canonical left action on itself and the right action induced by ff. Namely this bimdoule correspondonds to the map

Δ:C(G)C(G×G)C(G)C(G) \Delta \colon C(G) \to C(G \times G) \simeq C(G) \otimes C(G)

given on ϕC(G)\phi \in C(G) and g 1,g 2Gg_1, g_2 \in G by

Δϕ:(g 1,g 2)ϕ. \Delta \phi \colon (g_1, g_2) \mapsto \phi \,.

In summary this means that (for GG a finite group)

  1. if we regard BG\mathbf{B}G as presented as a double groupoid constant on BG\mathbf{B}G, then the corresponding groupoid convolution sesquialgebra (basis for a 3-module) is the convolution algebra of GG;

  2. if instead we regard BG\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 GG.


Steenrod operations on coTor groups

For Γ\Gamma a suitable commutative Hopf algebroid and N 1,N 2N_1, N_2 two Γ\Gamma-comodules, then the coTor groups CoTor Γ(N 1,N 2)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.

Generalized dual Steenrod algebra

For EE a suitable E-infinity ring spectrum, its homotopy groups π (E)\pi_\bullet(E) and generalized homology E (E)E_\bullet(E) form a Hopf algebroid of spectra, the dual EE-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 RR be an E-∞ ring and let AA an E-∞ algebra over RR. The self-generalized homology A R(A)A^R_\bullet(A) is naturally a module over the cohomology ring A A_\bullet via applying the homotopy groups \infty-functor π \pi_\bullet to the canonical inclusion

AARRARA. A \stackrel{\simeq}{\rightarrow} A \underset{R}{\wedge} R \stackrel{}{\rightarrow} A \underset{R}{\wedge} A \,.

Let RR be an E-∞ ring and let AA an E-∞ algebra over RR. If the the A A_\bullet-module A R(A)A^R_\bullet(A) of lemma is a flat module, then

  1. (A ,A (A))(A_\bullet, A_\bullet(A)) is a Hopf algebroid over R bullletR_\bulllet;

  2. A R(X)A^R_\bullet(X) is a left A R(A)A^R_\bullet(A)-module for every RR-∞-module XX.

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:

  • Georges Maltsiniotis, Groupoïdes quantiques, Comptes R

    Rendus Acad. Sci. Paris 314, pp. 249-252 (1992) ps

  • G. Maltsiniotis, Quasi-groupoïdes quantiques (travail en commun avec A. Bruguières), C.R. Acad. Sci. Paris 319, pp. 933-936 (1994) ps

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

For the relation to groupoid convolution algebras see also at groupoid convolution algebra – References – Convolution Hopf algebroids.

category: algebra

Last revised on February 28, 2021 at 16:41:37. See the history of this page for a list of all contributions to it.