Article
Full entry |
PDF
(0.6 MB)
Feedback
PDF
(0.6 MB)
Feedback
Keywords:
operad; skew monoidal category; operadic category
operad; skew monoidal category; operadic category
Summary:
I describe a generalization of the notion of operadic category due to Batanin and Markl. For each such operadic category I describe a skew monoidal category of collections, such that a monoid in this skew monoidal category is precisely an operad over the operadic category. In fact I describe two skew monoidal categories with this property. The first has the feature that the operadic category can be recovered from the skew monoidal category of collections; the second has the feature that the right unit constraint is invertible. In the case of the operadic category $\mathcal{S}$ of finite sets and functions, for which an operad is just a symmetric operad in the usual sense, the first skew monoidal category has underlying category [$\Bbb N, \bf Set$], and the second is the usual monoidal category of collections [$\Bbb P, \bf Set$] with the substitution monoidal structure.
References:
[1] Andrianopoulos, Jim: Units of skew monoidal categories and skew monoidales in Span. Master’s thesis, Macquarie University
[2] Barwick, C.: From operator categories to topological operads. Geometry and Topology 22(4):1893–1959
[3] Batanin, M. A.: Monoidal globular categories as a natural environment for the theory of weak n-categories. Adv. Math., 136(1):39–103
[4] Batanin, Michael, Markl, Martin: Operadic categories and duoidal Deligne’s conjecture. Adv. Math., 285:1630–1687
[5] Boardman, J. M., Vogt, R. M.: Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Mathematics, Vol. 347. Springer-Verlag, Berlin-New York
[6] Getzler, E., Kapranov, M. M.: Modular operads. Compositio Math., 110(1):65–126
[7] Hermida, Claudio: From coherent structures to universal properties. J. Pure Appl. Algebra, 165(1):7–61
[8] Kelly, G. M.: On the operads of J. P. May. Repr. Theory Appl. Categ., 13:1–13 (electronic)
[9] Lack, Stephen, Street, Ross: Skew monoidales, skew warpings and quantum categories. Theory Appl. Categ., 26:385–402
[10] Lack, Stephen, Street, Ross: On monads and warpings. Cah. Topol. Géom. Différ. Catég., 55(4):244–266
[11] Leinster, Tom: Higher operads, higher categories, volume 298 of LMS Lecture Note Series. Cambridge University Press, Cambridge
[12] Loday, Jean-Louis, Stasheff, James, Voronov, Alexander: Operads: Proceedings of Renaissance Conferences, volume 202 of Contemporary Mathematics. AMS
[13] May, J. P.: The geometry of iterated loop spaces. Springer-Verlag, Berlin. Lectures Notes in Mathematics, Vol. 271
[14] Szlachányi, Kornél: Skew-monoidal categories and bialgebroids. Adv. Math., 231(3-4):1694–1730
Search
Partner of
