Homotopy cooperads

V. V. Lyubashenko

doi:10.15330/ms.44.2.119-160

The theory of 2-monads is used as a ground to study non-symmetric cooperads. We give a new definition of homotopy cooperads. Ordinary and homotopy cooperads are placed in lax Cat-operads which are lax algebras over the free-operad strict 2-monad. We give an example of a homotopy cooperad cofree with respect to ordinary cooperads.

1. R. Blackwell, G.M. Kelly, A.J. Power Two-dimensional monad theory, J. Pure Appl. Algebra, 59 (1989), no.1, 1–41.

2. Yu. Bespalov, V.V. Lyubashenko, O. Manzyuk, Pretriangulated A1-categories, Proceedings of the Institute of Mathematics of NAS of Ukraine. Mathematics and its Applications, V.76, Institute of Mathematics of NAS of Ukraine, Kyiv, 2008.

3. D. Borisov, Yu.I. Manin, Generalized operads and their inner cohomomorphisms, Geometry and dynamics of groups and spaces, Progr. Math., V.265, Birkhauser, Basel, 2008, 247–308, http://arXiv.org/abs /math.CT/0609748.

4. F. Borceux, Handbook of categorical algebra. 1, Encyclopedia of Mathematics and its Applications, V.50, Cambridge University Press, 1994, Basic category theory.

5. R.E. Borcherds, Vertex algebras, Topological field theory, primitive forms and related topics (Kyoto, 1996) (M. Kashiwara, A. Matsuo, K. Saito, and I. Satake, eds.), Progr. Math., V.160, BirkhЁauser Boston, Boston, MA, 1998, 35–77, http://arXiv.org/abs/q-alg/9706008.

6. V.L. Ginzburg, M.M. Kapranov, Koszul duality for operads, Duke Math. J., 76 (1994), no.1, 203–272, http://arXiv.org/abs/0709.1228.

7. C. Hermida, From coherent structures to universal properties, J. Pure Appl. Algebra, 165 (2001), no.1, 7–61, http://arXiv.org/abs/math.CT/0006161.

8. M. Kontsevich, Yu.I. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys., 164 (1994), 525–562, http://arXiv.org/abs/hep-th/9402147.

9. S. Lack, A 2-categories companion, Towards higher categories, IMA Vol. Math. Appl., V.152, Springer, New York, New York, 2010, 105–191, http://arXiv.org/abs/math.CT/0702535.

10. T. Leinster, Up-to-homotopy monoids, 1999, http://arXiv.org/abs/math/9912084.

11. T. Leinster, Homotopy algebras for operads, 2000, http://arXiv.org/abs/math/0002180.

12. T. Leinster, Higher operads, higher categories, London Math. Soc. Lect. Notes Series, Cambridge University Press, Boston, Basel, Berlin, 2003, http://arXiv.org/abs/math/0305049.

13. J.-L. Loday, B. Vallette, Algebraic operads, Grundlehren der Mathematischen Wissenschaften, V.346, Springer, Heidelberg, 2012.

14. V.V. Lyubashenko, Curved homotopy coalgebras, 2014, http://arXiv.org/abs/1402.0408.

15. V.V. Lyubashenko, Curved cooperads and homotopy unital A1-algebras, 2014, http://arXiv.org/ abs/1403.3644.

16. S. Mac Lane, Categories for the working mathematician, GTM, V.5, Springer-Verlag, New York, 1971, 1988.

17. M. Markl, S. Shnider, J.D. Stasheff, Operads in algebra, topology and physics, Mathematical Surveys and Monographs, V.96, AMS, Providence, RI, 2002.

18. Ya.S. Soibelman, Meromorphic braided category arising in quantum affine algebras, Internat. Math. Res. Notices, 19 (1999), 1067–1079, http://arXiv.org/abs/math.QA/9901003.

19. R.H. Street, Fibrations and Yoneda’s lemma in a 2-category, Lect. Notes in Math., 420 (1974), 104–133.

	Homotopy cooperads
Author	V. V. Lyubashenko lub@imath.kiev.ua Institute of Mathematics NASU of Ukraine
Abstract	The theory of 2-monads is used as a ground to study non-symmetric cooperads. We give a new definition of homotopy cooperads. Ordinary and homotopy cooperads are placed in lax Cat-operads which are lax algebras over the free-operad strict 2-monad. We give an example of a homotopy cooperad cofree with respect to ordinary cooperads.
Keywords	cooperads; 2-monads; morphism classifiers
DOI	doi:10.15330/ms.44.2.119-160
Reference	1. R. Blackwell, G.M. Kelly, A.J. Power Two-dimensional monad theory, J. Pure Appl. Algebra, 59 (1989), no.1, 1–41. 2. Yu. Bespalov, V.V. Lyubashenko, O. Manzyuk, Pretriangulated A1-categories, Proceedings of the Institute of Mathematics of NAS of Ukraine. Mathematics and its Applications, V.76, Institute of Mathematics of NAS of Ukraine, Kyiv, 2008. 3. D. Borisov, Yu.I. Manin, Generalized operads and their inner cohomomorphisms, Geometry and dynamics of groups and spaces, Progr. Math., V.265, Birkhauser, Basel, 2008, 247–308, http://arXiv.org/abs /math.CT/0609748. 4. F. Borceux, Handbook of categorical algebra. 1, Encyclopedia of Mathematics and its Applications, V.50, Cambridge University Press, 1994, Basic category theory. 5. R.E. Borcherds, Vertex algebras, Topological field theory, primitive forms and related topics (Kyoto, 1996) (M. Kashiwara, A. Matsuo, K. Saito, and I. Satake, eds.), Progr. Math., V.160, BirkhЁauser Boston, Boston, MA, 1998, 35–77, http://arXiv.org/abs/q-alg/9706008. 6. V.L. Ginzburg, M.M. Kapranov, Koszul duality for operads, Duke Math. J., 76 (1994), no.1, 203–272, http://arXiv.org/abs/0709.1228. 7. C. Hermida, From coherent structures to universal properties, J. Pure Appl. Algebra, 165 (2001), no.1, 7–61, http://arXiv.org/abs/math.CT/0006161. 8. M. Kontsevich, Yu.I. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys., 164 (1994), 525–562, http://arXiv.org/abs/hep-th/9402147. 9. S. Lack, A 2-categories companion, Towards higher categories, IMA Vol. Math. Appl., V.152, Springer, New York, New York, 2010, 105–191, http://arXiv.org/abs/math.CT/0702535. 10. T. Leinster, Up-to-homotopy monoids, 1999, http://arXiv.org/abs/math/9912084. 11. T. Leinster, Homotopy algebras for operads, 2000, http://arXiv.org/abs/math/0002180. 12. T. Leinster, Higher operads, higher categories, London Math. Soc. Lect. Notes Series, Cambridge University Press, Boston, Basel, Berlin, 2003, http://arXiv.org/abs/math/0305049. 13. J.-L. Loday, B. Vallette, Algebraic operads, Grundlehren der Mathematischen Wissenschaften, V.346, Springer, Heidelberg, 2012. 14. V.V. Lyubashenko, Curved homotopy coalgebras, 2014, http://arXiv.org/abs/1402.0408. 15. V.V. Lyubashenko, Curved cooperads and homotopy unital A1-algebras, 2014, http://arXiv.org/ abs/1403.3644. 16. S. Mac Lane, Categories for the working mathematician, GTM, V.5, Springer-Verlag, New York, 1971, 1988. 17. M. Markl, S. Shnider, J.D. Stasheff, Operads in algebra, topology and physics, Mathematical Surveys and Monographs, V.96, AMS, Providence, RI, 2002. 18. Ya.S. Soibelman, Meromorphic braided category arising in quantum affine algebras, Internat. Math. Res. Notices, 19 (1999), 1067–1079, http://arXiv.org/abs/math.QA/9901003. 19. R.H. Street, Fibrations and Yoneda’s lemma in a 2-category, Lect. Notes in Math., 420 (1974), 104–133.
Pages	119-160
Volume	44
Issue	2
Year	2015
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html