Abstract
We describe the category of homotopy coalgebras, concentrating on properties of relatively cofree homotopy coalgebras, morphisms and coderivations from an ordinary coalgebra to a relatively cofree homotopy coalgebra, morphisms and coderivations between coalgebras of latter type. Cobar- and bar-constructions between counit-complemented curved coalgebras, unit-complemented curved algebras and curved homotopy coalgebras are described. Using twisting cochains an adjunction between cobar- and bar-constructions is derived under additional assumptions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bespalov, Y., Lyubashenko, V.V., Manzyuk, O.: Pretriangulated A ∞-categories, Proceedings of the Institute of Mathematics of NAS of Ukraine. In: Mathematics and its applications, vol. 76. Institute of mathematics of NAS of Ukraine, Kyiv (2008). http://www.math.ksu.edu/~lub/papers.html
Fukaya, K., Oh, Y.G., Ohta, H., Ono, K.: Lagrangian intersection Floer theory: anomaly and obstruction. In: AMS/IP studies in advanced mathematics series, vol. 46. American Mathematical Society (2009)
Hirsh, J., Millès, J.: Curved Koszul duality theory. Math. Ann. 354(4), 1465–1520 (2012). arXiv:1008.5368, doi:10.1007/s00208-011-0766-9
Lefèvre-Hasegawa, K.: Sur les A ∞-catégories. Ph.D. thesis, Université Paris 7, U.F.R. de Mathématiques (2003). arXiv:math.CT/0310337
Leinster, T.: Up-to-homotopy monoids (1999). arXiv:math/9912084
Leinster, T.: Homotopy algebras for operads (2000). arXiv:math/0002180
Leinster, T.: Higher operads, higher categories. London mathematical society lecture notes series. Cambridge University Press, Boston (2003). arXiv:math/0305049
Lyubashenko, V.V.: Bar and cobar constructions for curved algebras and coalgebras. Matematychni Studii 40(2), 115–131 (2013). arXiv:1311.7681
Lyubashenko, V.V.: Curved Cooperads and Homotopy Unital A ∞-algebras. In: Mathematical studies monograph series, vol. 17. Lviv, VNTL Publishers (2015)
Mac Lane, S.: Categories for the working mathematician, GTM, vol. 5. Springer, New York (1971)
Positselski, L.: Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence. Memoirs Amer. Math. Soc. 212(996), vi+133 (2011). arXiv:0905.2621
Positselski, L.: Weakly curved A ∞-algebras over a topological local ring (2012). arXiv:1202.2697
Pridham, J.P.: The homotopy theory of strong homotopy algebras and bialgebras. Homology, Homotopy Appl. 12(2), 39–108 (2010). arXiv:0908.0116
Verdier, J.L.: Des cat’egories d’eriv’ees des cat’egories ab’eliennes. Astérisque 239, xii+253 (1996). With a preface by Luc Illusie, Edited and with a note by Georges Maltsiniotis
Author information
Authors and Affiliations
Corresponding author
Additional information
To the memory of Ukrainian mathematician Yuriy Victorovych Bodnarchuk
Rights and permissions
About this article
Cite this article
Lyubashenko, V. Curved Homotopy Coalgebras. Appl Categor Struct 25, 991–1036 (2017). https://doi.org/10.1007/s10485-016-9440-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-016-9440-4