{
  "annotation_id": "6fabb70c-98b7-41b7-ba65-2d7ec1cdd48f",
  "date_created": "2026-02-17T05:53:16.234000Z",
  "date_modified": "2026-02-17T05:53:16.234000Z",
  "file_hash": "5f54fae32e6e5da752de5d9e416ba5285bc7dd12dbfe4a0d2ef93be62e719cae",
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  "record": {
    "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.\n  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.\n  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.\n  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.\n  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.",
    "arxiv_id": "2601.08802",
    "authors": [
      "William Balderrama",
      "Piotr Pstr\u0105gowski"
    ],
    "categories": [
      "math.AT",
      "math.CT"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "Unstable synthetic deformations I: Malcev theories",
    "url": "https://arxiv.org/abs/2601.08802",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "9b0af0f5-786a-4ea4-b01c-4cd1902c6df3",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}