Abstract
One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads (Batanin, Adv Math 136:39–103, 1998) to this task. We present a general construction of a tensor product on the category of n-globular sets from any normalised (n + 1)-operad A, in such a way that the algebras for A may be recaptured as enriched categories for the induced tensor product. This is an important step in reconciling the globular and simplicial approaches to higher category theory, because in the simplicial approaches one proceeds inductively following the idea that a weak (n + 1)-category is something like a category enriched in weak n-categories. In this paper we reveal how such an intuition may be formulated in terms of globular operads.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Batanin, M.: Monoidal globular categories as a natural environment for the theory of weak n-categories. Adv. Math. 136, 39–103 (1998)
Batanin, M.: The Eckmann-Hilton argument and higher operads. Adv. Math. 217, 334–385 (2008)
Berger, C.: Iterated wreath product of the simplex category and iterated loop spaces. Adv. Math. 213, 230–270 (2007)
Carboni, A., Lack, S., Walters, R.F.C.: Introduction to extensive and distributive categories. J. Pure Appl. Algebra 84, 145–158 (1993)
Crans, S.: A tensor product for gray categories. Theory Appl. Categ. 5, 12–69 (1999)
Day, B., Street, R.: Lax monoids, pseudo-operads, and convolution. Contemp. Math. 318, 75–96 (2003) (Diagrammatic Morphisms and Applications)
Gordon, R., Power, A.J., Street, R.: Coherence for tricategories. Mem. Amer. Math. Soc. AMS 117 (1995)
Kelly, G.M.: Basic Concepts of Enriched Category Theory, LMS Lecture Note Series, vol. 64. Cambridge University Press, Cambridge (1982)
Kelly, G.M.: On the operads of J.P. May. TAC Reprints Series 13 (2005)
Lack, S.: Codescent objects and coherence. J. Pure Appl. Algebra 175, 223–241 (2002)
Leinster, T.: Higher Operads, Higher Categories. Lecture Note Series. London Mathematical Society, London (2003)
McClure, J., Smith, J.: Cosimplicial objects and little n-cubes I. Amer. J. Math 126, 1109–1153 (2004)
Weber, M.: Generic morphisms, parametric representations, and weakly cartesian monads. Theory Appl. Categ. 13, 191–234 (2004)
Weber, M.: Familial 2-functors and parametric right adjoints. Theory Appl. Categ. 18, 665–732 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
To the memory of Max Kelly.
Rights and permissions
About this article
Cite this article
Batanin, M., Weber, M. Algebras of Higher Operads as Enriched Categories. Appl Categor Struct 19, 93–135 (2011). https://doi.org/10.1007/s10485-008-9179-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-008-9179-7