Unstable synthetic deformations I: Malcev theories
William Balderrama, Piotr Pstrągowski
TL;DR
This paper establishes a foundation for unstable deformations of $ ext{∞}$-categories using infinitary Malcev theories, defining theories, loop theories, and derived functors, and proving that the $ ext{∞}$-category of models is the free cocompletion under geometric realizations. It develops a robust framework connecting Kan conditions, Malcev operations, and universally Kan objects, and introduces a spiral/Goerss–Hopkins tower to organize nonabelian deformations. By analyzing discrete, bounded, and additive cases, it relates infinitary theories to classical homotopy theory and animation, and shows how derived functors act in this setting, including notions of weak left exactness. The work provides a flexible algebraic approach to model unstable spectral sequences and obstruction theories, with concrete instantiations such as synthetic spaces, synthetic $ ext{E}_k$-algebras, and filtered/module-like deformations, and sets the stage for the subsequent USD2/USD3 developments and applications to synthetic spectra and Goerss–Hopkins-type obstructions. Overall, the framework yields a cohesion between higher universal algebra and unstable homotopy theory, enabling systematic construction and analysis of deformations across a broad class of $ ext{∞}$-categories.
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.