Noncommutative Mackey functors and Hopf-cyclic homology.
Duality of Hopf-cyclic homology and cohomology is described by using Mackey functors. The description is based on the fact that for every natural equivalence \(\omega\colon F_1|_{\mathcal C}\to F_2|_{\mathcal C}\), \({\mathcal C}={\mathcal C}_1\cap{\mathcal C}_2\), there exist functors \(F_i'\colon{\mathcal C}'\to{\mathcal A}\), \({\mathcal C}'= {\mathcal C}_1\cup_{\mathcal C}{\mathcal C}_2\) and a natural equivalence \(\omega'\colon F_1'\to F_2'\) such that \(F_i'|_{\mathcal C_i}=F_i\) and \(\omega'|_{\mathcal C}=\omega\) (Prop.1.2). For this purpose, first the categories \({\mathcal F}(\text{Ass})\) and \({\mathcal F}_+(\text{Ass})\), whose objects are finite sets \([n]=\{0,\dots,n\}\) and finite sets and the empty set \(\{-1\}\) with orderings, and the categories \(\Delta\) and \(\Delta_+\) whose objects are the same as that of \({\mathcal F}(\text{Ass})\) and \({\mathcal F}_+(\text{Ass})\) and morphisms are compositions of injections and surjections, are introduced (\S2.2). Then, as subcategories of \({\mathcal F}(\text{Ass})\), \({\mathcal F}_\bullet(\text{Ass})\) and \({\mathcal F}^\bullet(\text{Ass})\) are defined. They have the same objects as \({\mathcal F}(\text{Ass})\), but objects are based maps \(f\colon[n]\to[m]\); \(f(0)=0\) or \(f(n)=m\). For a left (right) \(A\)-module \(M\), the functors \({\mathbf M}^\bullet\) and \({\mathbf M}_\bullet\) are defined by \[ {\mathbf M}^\bullet[n]=A^{\otimes n}\otimes_kM,\quad{\mathbf M}_\bullet[n]=M\otimes_kA^{\otimes n}. \] Let \(J\colon\Delta^{\text{op}}_+\to\Delta\) be the morphism \(J[n]= [n+1]\), then Hochschild complexes for an \(A\)-module and for a \(C\)-bicomplex, are \({\mathbf M}J|_{\Delta^{\text{op}}}\) and \({\mathbf M}J^{\text{op}}|_\Delta\) (\S2.4). The category \(\mathcal{QF}(\text{Ass})\) is a particular case of the generalized Quillen \(\mathcal Q\)-construction [\textit{B. L. Feigin} and \textit{B. L. Tsygan}, K-theory, arithmetic and geometry, Semin., Moscow Univ. 1984-86, Lect. Notes Math. 1289, 67-209 (1987; Zbl 0635.18008) and \textit{T. Pirashvili} [Cah. Topol. Géom. Différ. Catég. 43, No. 3, 221-239 (2002; Zbl 1057.18005)]. Functors defined on \(\mathcal{QF}(\text{Ass})\) are called (noncommutative) Mackey functors (\S2.7). If \(H\) is a monoid, then it induces a functor \({\mathbf H}\colon\mathcal{QF}(\text{Ass})\to\mathcal Set\) such that \({\mathbf H}([n])=H^{n+1}\). \({\mathbf H}J_0\) is a classifying space of \(H\). If \(H\) is a group, then there is a cyclic structure on \({\mathbf H}J_0\) (\S2.9., cf. \textit{D. Burghelea} and \textit{Z. Fiedorowicz} [Topology 25, 303-317 (1986; Zbl 0639.55003)]). Let \(B\) be a \(k\)-bialgebra with unit \(e\) and counit \(\varepsilon\), and \(M\) a left \(B\)-module. Then by Prop.2.1, functors \({\mathbf M}_i\colon{\mathcal C}_i\to k\text{-Vect}\), \(i=1,2\), are defined by \({\mathbf M}_1[n+1]=B^{\otimes n}\otimes_kM\) and \({\mathbf M}_2[n+1]=M\otimes_kB^{\otimes n}\), where \[ {\mathcal C}_1={\mathcal F}^\bullet(\text{Ass})\cap_{{\mathcal F}_+(\text{Ass})}\mathcal{QF}(\text{Ass}),\quad {\mathcal C}_2={\mathcal F}_\bullet(\text{Ass})^{\text{op}}\cap_{{\mathcal F}_+(\text{Ass})^{\text{op}}}\mathcal{QF}(\text{Ass}). \] After these preliminaries, existence of functors \(J_1\colon\Delta^{\text{op}}\to{\mathcal C}_1\), \(J_2\colon\Delta\to{\mathcal C}_2\) such that \({\mathbf M}_1J_1\) defines the Hochschild complex for a \(B\)-bimodule with the right action given by the counit of \(B\) and \({\mathbf M}_2J_2\) defines the Hochschild complex of a \(B\)-module \(M\) with the left action given by the unit of \(B\), is shown (Prop.3.1). Then, for a functor \({\mathcal M}\colon\widetilde{\mathcal{QF}(\text{Ass})}\to{\mathcal A}\); \[ \mu_n={\mathcal M}(s_1s^*_2s_2s_3^*\cdots s_{n-1}s^*_n)\colon{\mathcal M}[n+1]\to{\mathcal M}[n+1], \] where \(s_i\colon[n]\to[n-1]\) is the surjection \(s_i(i)=s_i(i+1)\), is an isomorphism, existence of functors \({\mathcal M}_i'\colon\Delta\cup_{\Delta\bullet} J'(\Delta^{\text{op}})\to{\mathcal A}\), \(i=1,2\), and a natural equivalence \(\omega\colon{\mathcal M}_1'\to{\mathcal M}_2'\) such that \[ {\mathcal M}_i'([n])={\mathcal M}[n+1],\quad{\mathcal M}_1'J'={\mathcal M}J_1,\quad{\mathcal M}_2'|_\Delta={\mathcal M}J_2,\quad\omega[n]=\mu_n, \] is proved (Th.3.4). In \S4, the last \S, conditions which imply that the functors \({\mathcal M}_i'\), \(i=1,2\), can be defiend on \(\Delta{\mathbf C}\) are treated and shown if \(M\) is a stable anti-Yetter-Drinfeld \(B\)-module (cf. \textit{P. M. Hajac}, \textit{M. Khalkhali}, \textit{B. Rangipour} and \textit{Y. Sommerhäuser} [C. R., Math., Acad. Sci. Paris 338, No. 8, 587-590 (2004; Zbl 1060.16037)]), \({\mathcal M}_i'\), \(i=1,2\), are defined on \(\Delta{\mathbf C}\) (Prop.4.5).
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