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\section*{Hopf algebroid over a commutative base}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{higher_groupoid_convolution_algebras_and_nvector_spacesnmodules}{Higher groupoid convolution algebras and n-vector spaces/n-modules}\dotfill \pageref*{higher_groupoid_convolution_algebras_and_nvector_spacesnmodules} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{steenrod_operations_on_cotor_groups}{Steenrod operations on coTor groups}\dotfill \pageref*{steenrod_operations_on_cotor_groups} \linebreak
\noindent\hyperlink{GeneralizedDualSteenrodAlgebra}{Generalized dual Steenrod algebra}\dotfill \pageref*{GeneralizedDualSteenrodAlgebra} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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]].

\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Just as for [[groups]] and their [[Hopf algebras]], there are \emph{two} ways to assign a \textbf{Hopf algebroid over a commutative base algebra} to a [[groupoid]] $\mathcal{G}_\bullet$:

\begin{enumerate}%
\item ([[commutative Hopf algebroids]]) with \textbf{commutative but non-co-commutative total algebra}: Form the [[commutative algebra|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);


\item with \textbf{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 $C(\mathcal{G}_0)$.



\end{enumerate}
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 [[dual equivalence|antiequivalence of categories]]. Commutative Hopf algebroids are useful also in a version in [[brave new algebra]] (the work of John Rognes).

\hypertarget{higher_groupoid_convolution_algebras_and_nvector_spacesnmodules}{}\subsection*{{Higher groupoid convolution algebras and n-vector spaces/n-modules}}\label{higher_groupoid_convolution_algebras_and_nvector_spacesnmodules}

\begin{quote}%
under construction


\end{quote}
We discuss here a natural generalization of the notion of [[groupoid convolution algebras]] to [[higher algebra|higher algebras]] for [[n-groupoid|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 [[discrete ∞-groupoid|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

\begin{remark}
\label{}\hypertarget{}{}
By the discussion at \emph{[[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

\begin{displaymath}
C \;\colon\; Grpd \to 2 Mod
  \,.
\end{displaymath}
There is then the following systematic refinement of this to higher groupoids and higher algebra: by the discussion at \emph{[[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 \ref{DoubleGroupoid2AlgebraOfDeloopingOfFiniteGroup} below.

\end{remark}
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-[[coskeleton|coskeletal]] as a [[simplicial object]] in [[Grpd]].

\begin{remark}
\label{}\hypertarget{}{}
Given a [[groupoid object in an (∞,1)-category|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:

\begin{itemize}%
\item groupoid convolution algebras $C(\mathcal{G}_0)$ and $C(\mathcal{G}_1)$,


\item 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 $\partial_1 \colon \mathcal{G}_1 \underset{\mathcal{G}_0}{\times} \mathcal{G}_1 \to \mathcal{G}_1$.



\end{itemize}
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 \textbf{double groupoid convolution 2-algebra} of $\mathcal{G}_\bullet$.

(Here we make invariant sense of the [[tensor product]] by evaluating on a [[Reedy model structure|Reedy fibrant]] representative.)

\end{remark}
\begin{example}
\label{DoubleGroupoid2AlgebraOfDeloopingOfFiniteGroup}\hypertarget{DoubleGroupoid2AlgebraOfDeloopingOfFiniteGroup}{}
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]]:

\begin{enumerate}%
\item $\underset{\longrightarrow}{\lim}(\cdots \mathbf{B}G \stackrel{\overset{id}{\to}}{\underset{id}{\to}} \mathbf{B}G) \simeq \mathbf{B}G$;


\item $\underset{\longrightarrow}{\lim}(\cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} *) \simeq \mathbf{B}G$.



\end{enumerate}
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 \emph{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

\begin{displaymath}
\Delta \colon C(G) \to C(G \times G) \simeq C(G) \otimes C(G)
\end{displaymath}
given on $\phi \in C(G)$ and $g_1, g_2 \in G$ by

\begin{displaymath}
\Delta \phi \colon (g_1, g_2) \mapsto \phi
  \,.
\end{displaymath}
In summary this means that (for $G$ a finite group)

\begin{enumerate}%
\item 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 [[n-module|3-module]]) is the convolution algebra of $G$;


\item 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 $G$.



\end{enumerate}
\end{example}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{steenrod_operations_on_cotor_groups}{}\subsubsection*{{Steenrod operations on coTor groups}}\label{steenrod_operations_on_cotor_groups}

For $\Gamma$ a suitable commutative Hopf algebroid and $N_1, N_2$ two $\Gamma$-[[comodules]], then the co[[Tor]] groups $CoTor_\Gamma(N_1, N_2)$ form a \emph{[[Steenrod algebra]]}. See there for details and citations.

For [[MU]] this is the content of the [[Landweber-Novikov theorem]].

\hypertarget{GeneralizedDualSteenrodAlgebra}{}\subsubsection*{{Generalized dual Steenrod algebra}}\label{GeneralizedDualSteenrodAlgebra}

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 \emph{\href{Steenrod+algebra#HopfAlgebraStructure}{Steenrod algebra -- Hopf algebroid structure}}.

The general statement is this:

\begin{lemma}
\label{SelfHomologyIsModuleOverCohomologyRing}\hypertarget{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

\begin{displaymath}
A \stackrel{\simeq}{\rightarrow}
  A \underset{R}{\wedge} R
  \stackrel{}{\rightarrow}
  A \underset{R}{\wedge} A
  \,.
\end{displaymath}
\end{lemma}
\begin{prop}
\label{}\hypertarget{}{}
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

\begin{enumerate}%
\item $(A_\bullet, A_\bullet(A))$ is a [[Hopf algebroid]] over $R_\bulllet$;


\item $A^R_\bullet(X)$ is a left $A^R_\bullet(A)$-module for every $R$-[[∞-module]] $X$.



\end{enumerate}
\end{prop}
This is due to (\hyperlink{BakerLazarev01}{Baker-Lazarev 01}), further discussed in (\hyperlink{BakerJeanneret02}{Baker-Jeanneret 02}) (there expressed in terms of the presentation by [[commutative monoids]] in [[symmetric spectra]]). A review is also in (\hyperlink{Ravenel}{Ravenel, chapter 2, prop. 2.2.8}).

\hypertarget{references}{}\subsection*{{References}}\label{references}

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 algebra|quasi-Hopf]] version:

\begin{itemize}%
\item [[Georges Maltsiniotis]], \emph{Groupo\"i{}des quantiques}, Comptes R Rendus Acad. Sci. Paris \textbf{314}, pp. 249-252 (1992) \href{http://people.math.jussieu.fr/~maltsin/ps/GRN-1.PS}{ps}
\item G. Maltsiniotis, \emph{Quasi-groupo\"i{}des quantiques} (travail en commun avec A. Brugui\`e{}res), C.R. Acad. Sci. Paris \textbf{319}, pp. 933-936 (1994) \href{http://people.math.jussieu.fr/~maltsin/ps/N-QSBIG.PS}{ps}

\end{itemize}
A [[Tannaka duality]]-type theorem relating certain subcategory of commutative Hopf algebroids to discrete groupoids is in

\begin{itemize}%
\item Laiachi EL Kaoutit, \emph{Representative functions on discrete groupoids and duality with Hopf algebroids}, \href{http://arxiv.org/abs/1311.3109}{arxiv/1311.3109}

\end{itemize}
For the relation to [[groupoid convolution algebras]] see also at \emph{\href{category%20algebra#ReferencesConvolutionHopfAlgebroids}{groupoid convolution algebra -- References -- Convolution Hopf algebroids}}.

\begin{itemize}%
\item [[Mark Hovey]], \emph{Morita theory for Hopf algebroids and presheaves of groupoids} (\href{http://arxiv.org/abs/math/0105137}{arXiv:math/0105137})

\end{itemize}
category: algebra

[[!redirects Hopf algebroids over a commutative base]]



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