The Inductive Coherator For Grothendieck Infinity Groupoids

TL;DR

The work tackles modeling Grothendieck ${\infty}$-groupoids by recasting higher-coherence data through monads and distributive laws. It generalizes distributive series of monads to an ${\mathbb N}$-indexed framework and introduces the notion of a completable series, whose associated pointed endofunctor has a colimit and lifts to a monad under suitable hypotheses. Two completable distributive series are constructed on the category of theories over $Θ_0^\mathrm{op}$ via factorization systems; the first yields an ${\infty}$-coherator $(\infty,0)$-coherator $IC$ through inductive weak enrichment, while the second produces a theory for strict ${\infty}$-groupoids. The approach relies on algebraic weak factorization systems and a Kelly-style Small Object argument, with appendices connecting the framework to epi-monads and explicit structure maps, highlighting the universal and inductive nature of the construction.

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 $Θ_0^\op$, 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 $Θ_0^\op$ 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 $Θ_0^\op$ to a theory for strict $\infty$-groupoids.