Iterated monoids and groups

Definition 5.6.1. Let \(C\) be an \(\infty \)-category with finite products. For \(n \geq 0\) we iteratively define the \(\infty \)-category of \(n\)-fold monoids in \(C\):

• For \(n = 0\) we define \(\Mon ^{(0)}(C) := C_*\);
• For \(n \geq 1\) we define \(\Mon ^{(n)}(C) := \Mon (\Mon ^{(n-1)}(C))\).

We similarly define the \(\infty \)-category of \(n\)-fold groups in \(C\) by \(\Grp ^{(0)}(C) = C_*\) and \(\Grp ^{(n)}(C) := \Grp (\Grp ^{(n-1)}(C))\).