Operads, props, and related structures are algebraic objects which encode multilinear structures. Operads arose in algebraic topology in order to address problems of homotopy coherence in the classification of loop spaces, but since the nineties they have found applications throughout mathematics, computer science and other applied sciences. As a general rule, operads classify familiar algebraic structures-associative algebras, commutative algebras, Lie algebras-and variants of these algebraic structures where operations might only hold up to coherent homotopy. A good introduction to operads is this article by Jim Stasheff, in the Notices of the AMS What is… an Operad? or this pair of blog posts by Tai-Danae Bradley: What is an Operad? Part 1 and Part 2. 

Props and properads classify more complicated algebraic structures like Lie bialgebras. They also arise in topological constructions like Segal's cobordism category and the (wheeled) props (aka circuit algebras) that describe virtual and welded tangles. Infinity, or weak, (pr)operads are (pr)operads up to coherent homotopy-meaning that the algebraic structure of the (pr)operad itself is only weakly associative (eg: infinity properads). These objects arise quite naturally whenever one wants to study topological objects which have an algebraic structure. Examples include configuration spaces, configuration categories, normalized cacti, or operads which have been subjected to completion.  

Preprints

  1. Dancso, Z., Halacheva, I., & Robertson, M. (2021b). A topological characterisation of the kashiwara-vergne groups. arXiv version.
  2. Bonatto, L. B., Chettih, S., Linton, A., Raynor, S., Robertson, M., & Wahl, N. (2020). An infinity operad of normalized cacti.  arXiv version.​

Publications


  1. Dancso, Z., Halacheva, I., & Robertson, M. (2021). Circuit algebras are wheeled props. J. Pure Appl. Algebra, 225(12), Paper No. 106767, 33. Retrieved from https://doi.org/10.1016/j.jpaa.2021.106767 doi: 10.1016/j.jpaa.2021.106767. arXiv version. ​​​​
  2. ​Hackney, P., Robertson, M., & Yau, D. (2020b). Modular operads and the nerve theorem. Adv. Math., 370, 107206, 39. Retrieved from https://doi.org/10.1016/j.aim.2020.107206 doi: 10.1016/j.aim.2020 .107206. arXiv version
  3. Hackney, P., Robertson, M., & Yau, D. (2020a). A graphical category for higher modular operads. Adv. Math., 365, 107044, 61. Retrieved from https://doi.org/10.1016/j.aim.2020.107044 doi: 10.1016/j.aim.2020.107044 
  4. Hackney, P., Robertson, M., & Yau, D. (2019). Higher cyclic operads. Algebr. Geom. Topol., 19(2), 863–940. Retrieved from https://doi.org/10.2140/agt.2019.19.863 doi: 10.2140/agt.2019.19.863. arXiv version. 
  5. Boavida de Brito, P., Horel, G., & Robertson, M. (2019). Operads of genus zero curves and the Grothendieck- Teichmüller group. Geom. Topol., 23(1), 299–346. Retrieved from https://doi.org/10.2140/gt.2019.23.299 doi: 10.2140/gt.2019.23.299. ​
  6. ​Hackney, P., Robertson, M., & Yau, D. (2018). On factorizations of graphical maps. Homology Homotopy Appl., 20(2), 217–238. Retrieved from https://doi.org/10.4310/HHA.2018.v20.n2.a11 doi: 10.4310/ HHA.2018.v20.n2.a11 arXiv version. 
  7. Hackney, P., & Robertson, M. (2018). Lecture notes on infinity-properads. In 2016 MATRIX annals (Vol. 1, pp. 351–374). Springer, Cham. ​
  8. Hackney, P., Robertson, M., & Yau, D. (2017b). A simplicial model for infinity properads. High. Struct., 1(1), 1–21. arXiv version.
  9. Hackney, P., Robertson, M., & Yau, D. (2017a). Shrinkability, relative left properness, and derived base change. New York J. Math., 23, 83–117. Retrieved from http://nyjm.albany.edu:8000/j/2017/23_83.html. arXiv version. 
  10. Hackney, P., & Robertson, M. (2017). The homotopy theory of simplicial props. Israel J. Math., 219(2), 835– 902. Retrieved from https://doi.org/10.1007/s11856-017-1500-4 doi: 10.1007/s11856-017-1500-4. arXiv version. 
  11. Hackney, P., Robertson, M., & Yau, D. (2016). Relative left properness of colored operads. Algebr. Geom. Topol., 16(5), 2691–2714. Retrieved from https://doi.org/10.2140/agt.2016.16.2691 doi: 10.2140/agt.2016.16.2691. arXiv version.
  12. Hackney, P., Robertson, M., & Yau, D. (2015). Infinity properads and infinity wheeled properads (Vol. 2147). Springer, Cham. Retrieved from https://doi.org/10.1007/978-3-319-20547-2 doi: 10.1007/ 978-3-319-20547-2.​
  13. Aponte Román, C. I., Livernet, M., Robertson, M., Whitehouse, S., & Ziegenhagen, S. (2015). Representations of derived A-infinity algebras. In Women in topology: collaborations in homotopy theory (Vol. 641, pp. 1–27). Amer. Math. Soc., Providence, RI. Retrieved from https://doi.org/10.1090/conm/641/12855 doi: 10.1090/conm/641/12855.  arXiv version. ​
  14. Bergner, J. E., & Robertson, M. (2015). Cluster categories for topologists. In Stacks and categories in geometry, topology, and algebra (Vol. 643, pp. 25–35). Amer. Math. Soc., Providence, RI. Retrieved from https://doi.org/10.1090/conm/643/12895 doi: 10.1090/conm/643/12895 ​
  15. Hackney, P., & Robertson, M. (2015). On the category of props. Appl. Categ. Structures, 23(4), 543–573. Retrieved from https://doi.org/10.1007/s10485-014-9369-4 doi: 10.1007/s10485-014-9369-4. arXiv version.​
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