Operads of higher transformations for globular sets and for higher magmas

Document Type : Research Paper


Department of Mathematics, Macquarie University, Sydney, Australia.


‎In this article we discuss examples of fractal $\omega$-operads‎. ‎Thus we show that there is an $\omega$-operadic approach to explain existence of‎ ‎the globular set of globular sets\footnote{Globular sets are also called $\omega$-graphs by the French School.}‎, ‎the reflexive globular set of reflexive globular sets‎, ‎the $\omega$-magma of $\omega$-magmas‎, ‎and also the reflexive $\omega$-magma of reflexive $\omega$-magmas‎. ‎Thus‎, ‎even though the existence of the‎ ‎globular set of globular sets is intuitively evident‎, ‎many other higher structures which \textit{fractality} are less evident‎, ‎could be described‎ ‎with the same technology‎, ‎using fractal $\omega$-operads‎. ‎We have in mind the non-trivial question of the existence of the‎ ‎weak $\omega$-category of the weak $\omega$-categories in the globular setting‎, ‎which is described in \cite{kach-ir3} with the same technology up to a contractibility‎ ‎hypothesis‎.


[1] J. Adamek and J. Rosicky, “Locally Presentable and Accessible Categories”, Cambridge
University Press, 1994.
[2] M. Batanin, Monoidal globular categories as a natural environment for the theory
of weak-n-categories, Adv. Math. 136 (1998), 39–103.
[3] F. Borceux, “Handbook of Categorical Algebra, Vol. 2”, Cambridge University
Press, 1994.
[4] C. Kachour, Definition algebrique des cellules non-strictes, Cah. Topol. Geom.
Differ. Categ. 1 (2008), pages 1–68.
[5] C. Kachour, Operadic definition of the non-strict cells, Cah. Topol. Geom. Differ.
Categ. 4 (2011), 1–48.
[6] C. Kachour, “Aspects of Globular Higher Category Theory”, Ph.D. Thesis, Macquarie
University, 2013.
[7] C. Kachour, Algebraic definition of weak (1, n)-categories, To appear in Theory
Appl. Categ. (2015).
[8] Camell Kachour, Operads of coendomorphisms and fractal operads for higher
structures, Categ. General Alg. Structures Appl. 3(1) (2015).
[9] C. Kachour, Steps toward the weak category of the weak categories in the globular
setting, To appear in Categ. General Alg. Structures Appl. 3(2) (2015).
[10] G.M. Kelly, A unified treatment of transfinite constructions for free algebras, free
monoids, colimits, associated sheaves, and so on, Bull. Aust. Math. Soc. 22 (1980),
[11] S. Lack, On the monadicity of finitary monads, J. Pure Appl. Algebra 140 (1999),
[12] L. Coppey and Ch. Lair, Le cons de theorie des esquisses, Universite Paris VII,
[13] T. Leinster, “Higher Operads, Higher Categories”, London Math. Soc. Lect. Note
Series, Cambridge University Press 298 (2004).
[14] M. Makkai and R. Pare, “Accessible Categories: The Foundations of Categorical
Model Theory”, American Mathematical Society, (1989).
[15] G. Maltsiniotis, Infini groupo¨ıdes non strictes, d’apr`es Grothendieck,
http://www.math.jussieu.fr/ maltsin/ps/infgrart.pdf (2007).
[16] J. Penon, Approche polygraphique des 1-categories non-strictes, Cah. Topol. Geom.
Differ. Categ. 1 (1999) 31–80.
[17] D. Tamarkin, What do DG categories form?, Compos. Math. 143(5) (2007), 1335–