Homotopy invariance of the cyclic homology of A ∞-algebras under homotopy equivalences of A ∞-algebras

In the present paper, the cyclic homology functor from the category of A ∞ {A_{\infty}} -algebras over any commutative unital ring K to the category of graded K-modules is constructed. Further, it is shown that this functor sends homotopy equivalences of A ∞ {A_{\infty}} -algebras into isomorphisms of graded modules. As a corollary, it is stated that the cyclic homology of an A ∞ {A_{\infty}} -algebra over any field is isomorphic to the cyclic homology of the A ∞ {A_{\infty}} -algebra of homologies for the source A ∞ {A_{\infty}} -algebra.

Mixed vs Stable Anti-Yetter-Drinfeld Contramodules

We examine the cyclic homology of the monoidal category of modules over a finite dimensional Hopf algebra, motivated by the need to demonstrate that there is a difference between the recently introduced mixed anti-Yetter-Drinfeld contramodules and the usual stable anti-Yetter-Drinfeld contramodules. Namely, we show that Sweedler's Hopf algebra provides an example where mixed complexes in the category of stable anti-Yetter-Drinfeld contramodules (previously studied) are not equivalent, as differential graded categories to the category of mixed anti-Yetter-Drinfeld contramodules (recently introduced).

Cyclic-Homology Chern–Weil Theory for Families of Principal Coactions

Viewing the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model, we construct the cyclic-homology Chern–Weil homomorphism. To realize the thus constructed Chern–Weil homomorphism as a Cartan model of the homomorphism tautologically induced by the classifying map on cohomology, we replace the unital subalgebra of coaction-invariants by its natural H-unital nilpotent extension (row extension). Although the row-extension algebra provides a drastically different model of the cyclic object, we prove that, for any row extension of any unital algebra over a commutative ring, the row-extension Hochschild complex and the usual Hochschild complex are chain homotopy equivalent. It is the discovery of an explicit homotopy formula that allows us to improve the homological quasi-isomorphism arguments of Loday and Wodzicki. We work with families of principal coactions, and instantiate our noncommutative Chern–Weil theory by computing the cotrace space and analyzing a dimension-drop-like effect in the spirit of Feng and Tsygan for the quantum-deformation family of the standard quantum Hopf fibrations.

Cyclic homology for bornological coarse spaces

The goal of the paper is to define Hochschild and cyclic homology for bornological coarse spaces, i.e., lax symmetric monoidal functors $${{\,\mathrm{\mathcal {X}HH}\,}}_{}^G$$ X HH G and $${{\,\mathrm{\mathcal {X}HC}\,}}_{}^G$$ X HC G from the category $$G\mathbf {BornCoarse}$$ G BornCoarse of equivariant bornological coarse spaces to the cocomplete stable $$\infty $$ ∞ -category $$\mathbf {Ch}_\infty $$ Ch ∞ of chain complexes reminiscent of the classical Hochschild and cyclic homology. We investigate relations to coarse algebraic K-theory $$\mathcal {X}K^G_{}$$ X K G and to coarse ordinary homology $${{\,\mathrm{\mathcal {X}H}\,}}^G$$ X H G by constructing a trace-like natural transformation $$\mathcal {X}K_{}^G\rightarrow {{\,\mathrm{\mathcal {X}H}\,}}^G$$ X K G → X H G that factors through coarse Hochschild (and cyclic) homology. We further compare the forget-control map for $${{\,\mathrm{\mathcal {X}HH}\,}}_{}^G$$ X HH G with the associated generalized assembly map.