Algebraic coherators, controlled theories, and Grothendieck realizations

Abstract

We introduce a construction of algebraic coherators for Grothendieck $\infty$-groupoids using the algebraic small object argument, replacing previous approaches we have used based on distributive series of monads with a more direct method for freely adjoining coherence data. Given a controlled theory, we define unreduced and reduced Grothendieck realizations, producing $\infty$-Lawvere theories and extending this construction functorially to connected diagrams of controlled theories. We apply this framework to construct globular models for monoidal $\infty$-groupoids, symmetric monoidal $\infty$-groupoids, coherent $\infty$-groups, and Picard $\infty$-groupoids. We define canonical semi-model structures on categories of models over $\infty$-Lawvere theories and formulate a generalized pushout conjecture that implies the existence of these semi-model structures and the Homotopy Hypothesis for Grothendieck $\infty$-groupoids.