Tuesday, August 23 2022 at 3:00 p.m. (sharp)

MSc Candidate: Mahmud Azam

Supervisor: Alexander Kupers

Thesis title: Semidirect Products of ∞–Operads

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We provide a construction of an $\infty$–operad from a functor $BG \to \Op_\infty$ encoding the action of a group $G$ on a given unital $\infty$–operad whose underlying $\infty$–category is a Kan complex. This construction, restricted to classical operads in $\Set$ viewed as $\infty$–operads, coincides with the semidirect product construction. Taking this as the definition of semidirect product of $\infty$–operads, we show that the action of $G$ on the given $\infty$–operad is equivalent to the trivial action if and only if the corresponding semidirect products

are equivalent. We then outline how one might generalize this result to operads in $\Top$

and use this to show that the semidirect product of the real version of the little $n$–disks operad with $SO(n – 1)$ or $SO(n – 2)$ for $n$ even or odd respectively corresponding to the usual action is equivalent to the semidirect product corresponding to the trivial action.

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A copy of the thesis can be found here:Operads

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