The Inductive Coherator For Grothendieck Infinity Groupoids

We extend the theory of distributive series of monads of \cite{EC1} by extending the definition to include an $\text{bN}$-indexed collection of monads. Under certain conditions, distributive series of monads will have a colimit in the category of pointed endofunctors. We define a \emph{completable} distributive series of monads to be a distributive series of monads whose induced pointed endofunctor, if it exists, lifts to a monad. We then construct factorization systems used to generate monads on the category of theories over ${\mathrm{\Theta }}_0^{\text{op}}$, in order to form two \emph{completable} distributive series of monads. The first completable distributive series of monads induces a monad that sends the identity theory over ${\mathrm{\Theta }}_0^{\text{op}}$ to an $(\mathrm{\infty },0)$-coherator whose inductive construction mimics inductive weak enrichment. The second completable distributive series of monads induces a monad that sends the identity theory over ${\mathrm{\Theta }}_0^{\text{op}}$ to a theory for strict $\mathrm{\infty }$-groupoids.