Unstable synthetic deformations I: Malcev theories

Abstract: This paper is the first in a series of articles devoted to the construction and study of synthetic deformations of $\infty$-categories in the unstable context: that is, deformations of $\infty$-categories that categorify spectral sequence or obstruction-theoretic information. This paper sets up the foundations of our study. We introduce and study various classes of $\infty$-categorical and infinitary algebraic theories. We establish many basic properties of the $\infty$-categories of the models of different classes of theories, as well as recognition theorems identifying the $\infty$-categories that arise this way. We give an intrinsic definition of a Malcev theory in higher universal algebra. We establish that the $\infty$-category of models of a Malcev theory may be characterized as freely adjoining geometric realizations to the theory. This leads to the notion of a derived functor between $\infty$-categories of models of Malcev theories, and we study the behavior of these derived functors with respect to connectivity and limits. We recall the notion of a loop theory and study in detail the interaction between functors and derived functors of $\infty$-categories of loop models and models, establishing that a large class of comonads on the $\infty$-category of loop models deform canonically to the $\infty$-category of all models. In the last part of the paper, we show that by considering the coalgebras for these deformed comonads over $\infty$-categories of models, one can recover various stable deformations considered in the literature, such as filtered models or Postnikov-complete synthetic spectra. We then expand on these results by constructing $\infty$-categories of synthetic spaces and synthetic $\mathbf{E}_k$-rings.