Little n-disks operad and n-operads
Organizers
Speaker
Michael Batanin
Time
Monday, March 30, 2026 4:00 PM - 5:00 PM
Venue
A3-1-301
Online
Zoom 559 700 6085
(BIMSA)
Abstract
n-operads, in general, describe n-category like structures (weak n-categories in particular). The arities of their spaces of operations are certain basic globular diagrams unlike classical operads whose arities of operations are natural numbers. There is, however, a closed connection between classical symmetric operads and n-operads given by a pair functors : desymmetrisation and its left adjoint called symmetrisation. In my talk I will define n-operads (more precisely certain important subcategory of n-operads) and this adjoint pair of functors.
Then I will show that there exists a particular nice cofibrant, contractible (!) topological n-operad GJ^n with the property that its symmetrisation is isomorphic to the celebrated Fulton-Macpherson operad fm^n obtained as a compactification of moduli space of configurations of points in R^n. This result shows that homotopically the little n-disks operad is the value of the left derived functor of symmetrisation on the terminal n-operad. This should be considered as a derived version of classical Eckman-Hilton argument or as a coherence theorem for E_n-algebras. Moreover, it implies that any weak (in appropriate sense) n-category which has only one object, one arrow , one 2-cells, ..., one (n-1)-cell is exactly the same as an algebra of the little n-disks operad. One consequence is a short proof of the Deligne conjecture on Hochschild cochains which I will provide if there is time.
Then I will show that there exists a particular nice cofibrant, contractible (!) topological n-operad GJ^n with the property that its symmetrisation is isomorphic to the celebrated Fulton-Macpherson operad fm^n obtained as a compactification of moduli space of configurations of points in R^n. This result shows that homotopically the little n-disks operad is the value of the left derived functor of symmetrisation on the terminal n-operad. This should be considered as a derived version of classical Eckman-Hilton argument or as a coherence theorem for E_n-algebras. Moreover, it implies that any weak (in appropriate sense) n-category which has only one object, one arrow , one 2-cells, ..., one (n-1)-cell is exactly the same as an algebra of the little n-disks operad. One consequence is a short proof of the Deligne conjecture on Hochschild cochains which I will provide if there is time.