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Disquisitions on Monoidal Categories and Operads
Disquisitions on Monoidal Categories and Operads
Little n-disks operad and n-operads
Little n-disks operad and n-operads
演讲者
Michael Batanin
时间
2026年03月30日 16:00 至 17:00
地点
A3-1-301
线上
Zoom 559 700 6085
(BIMSA)
摘要
$n$-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 $\mathbb{R}^n$. 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 $\mathbb{R}^n$. 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.