 ${{\mathcal C}}$. Section 5 is a postface, or a “discussion” (as they do in medical journals) – we discuss some of the further things one might (and should) do with cyclic bimodules, and how to correct some deficiencies of the theory developed in Sections 2 and 4.

Acknowledgements. {#acknowledgements. .unnumbered}
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In the course of this work, I have benefited greatly from discussions with A. Beilinson, E. Getzler, V. Ginzburg, A. Kuznetsov, N. Markarian, D. Tamarkin, and B. Tsygan. I am grateful to Northwestern Univeristy, where part of this work was done, and where some of the results were presented in seminars, with great indulgence from the audience towards the unfinished state they were in. And, last but not least, it is a great pleasure and a great opportunity to dedicate the paper to Yuri Ivanovich Manin on his birthday. Besides all the usual things, I would like to stress that it is the book [@GM1], – and [@GM2], to a lesser extent – which shaped the way we look at homological algebra today, at least “we” of my generation and of Moscow school. Without Manin’s decisive influence, this paper certainly would not have appeared (as in fact at least a half of the papers I ever wrote).

Recollection on cyclic homology.
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We start by recalling, extremely briefly, A. Connes’ approach to cyclic homology, which was originally introduced in [@C] (for detailed overviews, see e.g. [@L Section 6] or [@FT Appendix]; a brief but complete exposition using the same language and notation as in this paper can be found in [@Ka Section 1]).

Connes’ approach relies on the technique of homology of small categories. Fix a base field $k$. Recall that for every small category $\Gamma$, the category ${\operatorname{Fun}}(\Gamma,k)$ of functors from $\Gamma$ to the category $k{\operatorname{\it\!-Vect}}$ of $k$-vector spaces is an abelian category with enough projectives and enough injectives, with derived category ${{\mathcal D}}(\Gamma,k)$. For any object $E \in