2026/07/27 by Johnathon Taylor
#math.CT
In this paper, we introduce the notion of a controlled theory, originally developed in the author's thesis, as a structural tool for the study of higher categorical algebra. We define a notion of deformation for pros and controlled theories in a cartesian closed category. Furthermore, we show that deformations of controlled theories naturally produce Lawvere theories enriched over the same base category. We construct functorial one-dimensional categorifications and homotopifications of controlled theories, yielding Lawvere 2-theories and Lawvere theories enriched in simplicial sets, respectively. As an application, we obtain a new model for ∞-groups and construct a model of coherent group-like E_∞-spaces, which we will show in future work models infinite loop spaces.