# Twisting structures and morphisms up to strong homotopy

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## Abstract

We define twisted composition products of symmetric sequences via classifying morphisms rather than twisting cochains. Our approach allows us to establish an adjunction that simultaneously generalizes a classic one for algebras and coalgebras, and the bar-cobar adjunction for quadratic operads. The comonad associated to this adjunction turns out to be, in several cases, a standard Koszul construction. The associated Kleisli categories are the “strong homotopy” morphism categories. In an appendix, we study the co-ring associated to the canonical morphism of cooperads Open image in new window, which is exactly the two-sided Koszul resolution of the associative operad Open image in new window, also known as the Alexander-Whitney co-ring.

## Keywords

Composition product Classifying morphism Twisting cochain Kleisli category Strong homotopy morphism Koszul resolution## Notes

### Acknowledgements

This has proved to be a very long-term project, which has evolved significantly over the past nine years. Earlier versions of our approach to describing strongly homotopy morphisms via co-rings can be found on the arXiv [19]. Results from these earlier manuscripts, in particular concerning the Alexander-Whitney co-ring, which have already been applied in various articles and theses (e.g., [17, 18, 20, 21, 29] and [6]), are also stated and proved here. The first author would like to acknowledge the Mittag-Leffler Institute. The second author acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) (funding reference number 06133). Finally, the authors would like to acknowledge the support of the Midwest Topology Network.

## Supplementary material

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