The Inductive Coherator For Grothendieck Infinity Groupoids

TL;DR

This paper develops a new framework for constructing monads related to Grothendieck infinity groupoids using distributive series of monads, enabling inductive weak enrichment and strict infinity-groupoid theories.

Contribution

It extends the theory of distributive series of monads to include an -indexed collection, introducing the concept of completable series to construct monads for -coherators and strict -groupoids.

Findings

Constructed two -completable distributive series of monads.

Generated a monad for -coherators via inductive weak enrichment.

Produced a monad for strict -groupoids.

Abstract

We extend the theory of distributive series of monads of \cite{EC1} by extending the definition to include an $\bN$-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 $\Theta_0^\op$, 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 $\Theta_0^\op$ to an $(\infty,0)$-coherator whose inductive construction mimics inductive weak enrichment. The second completable distributive series of monads induces a monad that sends the identity theory over $\Theta_0^\op$ to a theory for strict $\infty$-groupoids.