Unstable synthetic deformations I: Malcev theories

TL;DR

This paper develops the foundations for synthetic deformations of $ $-categories in the unstable setting, introducing higher algebraic theories, Malcev theories, and their models, and connecting these to stable deformations and synthetic spectra.

Contribution

It introduces and studies higher universal algebra theories, especially Malcev theories, and their models, establishing their properties and connections to stable deformations and synthetic spectra.

Findings

Characterization of models of Malcev theories as freely adjoining geometric realizations

Development of derived functors between $ $-categories of models and their properties

Construction of $ $-categories of synthetic spaces and $ $-rings

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.