Hochschild homology and cohomology have a natural meaning via Tor and Ext groups; and they also have an infinity-categorical interpretation. The Hochschild complex for associative algebras has a remarkable quotient, the cyclic complex?; this construction is not as general as the mentioned construction, and it can not be generalized to algebras over an arbitrary operad. Instead there is an additional structure on an operad which enables one to produce an analogue of cyclic homology. However the long exact sequence of Connes which in the classical case involves cyclic homology and Hochschild homology, here involves the cyclic homology for the original cyclic operad but also the one for the Koszul dual operad? and the Hochschild. In the classical, associative case of course the operad and its Koszul dual coincide.

Other motivations

Cyclic operads also appear in TQFT-related constructions, often with more structure. See modular operads.

Literature

Ezra Getzler, M. M. Kapranov, Cyclic operads and cyclic homology, in “Geometry, topology and physics,“ International Press, Cambridge, MA, 1995, pp. 167-201 (pdf)

Jovana Obradović, Monoid-like definitions of cyclic operad, tac

Pierre-Louis Curien, Jovana Obradović, A formal language for cyclic operads, arXiv:1602.07502; Categorified cyclic operads, Appl. Categ. Struct. 28, 59–112 (2020) doi

Benjamin C. Ward, Maurer-Cartan elements and cyclic operads, J. Noncommut. Geom. 10:4 (2016) 1403–1464 doi

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