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Stable first order theories as simplicial profinite sets

Misha Gavrilovich

TL;DR

The paper reframes core model-theoretic notions—complete first-order theories, their models, and stability—within the category of simplicial profinite sets, using vibrant decalage properties to replace syntactic formulations. By modeling theories as symmetric simplicial objects $T_ullet$ with open/clopen maps and interpreting models as vibrant morphisms $M_ullet\to T_ullet$, it provides a purely category-theoretic lens on foundational FO concepts and their interrelations. It offers concrete examples (e.g., ACF$_0$, $ ext{R}$, $ ext{Vect}_k$, random graphs) and discusses how interpretations, reducts, and conservative extensions arise as morphisms, suggesting potential cohomological invariants via decalage. While the work is largely reformulatory, it proposes new avenues for universal constructions, stability theory, and even connections to deeper geometric ideas such as pro-finite compactifications and possible decalage cohomology, signaling potential cross-pollination with Knight’s approach and motivic phenomena.

Abstract

We rewrite simplicially the standard definitions of a complete first order theory, a model of it, and various characterisations of stability of a complete first order theory. In our reformulations the simplicial language replaces the standard definitions based on syntax, making them formally unnecessary. We view a complete first-order theory as a symmetric simplicial object in the category of profinite sets and open continuous maps, defined by the functor sending a finite set of variables into the Stone space of complete types in those variables. A model of a complete first-order theory is then a morphism from a representable simplicial set satisfying certain lifting properties reminiscent of, but weaker then, those in the definition of a fibration. The class of simplicial profinite sets corresponding to complete first order theories is characterised by the same lifting properties required of the map from the simplicial covering space (decalage) forgetting the extra degeneracy.

Stable first order theories as simplicial profinite sets

TL;DR

The paper reframes core model-theoretic notions—complete first-order theories, their models, and stability—within the category of simplicial profinite sets, using vibrant decalage properties to replace syntactic formulations. By modeling theories as symmetric simplicial objects with open/clopen maps and interpreting models as vibrant morphisms , it provides a purely category-theoretic lens on foundational FO concepts and their interrelations. It offers concrete examples (e.g., ACF, , , random graphs) and discusses how interpretations, reducts, and conservative extensions arise as morphisms, suggesting potential cohomological invariants via decalage. While the work is largely reformulatory, it proposes new avenues for universal constructions, stability theory, and even connections to deeper geometric ideas such as pro-finite compactifications and possible decalage cohomology, signaling potential cross-pollination with Knight’s approach and motivic phenomena.

Abstract

We rewrite simplicially the standard definitions of a complete first order theory, a model of it, and various characterisations of stability of a complete first order theory. In our reformulations the simplicial language replaces the standard definitions based on syntax, making them formally unnecessary. We view a complete first-order theory as a symmetric simplicial object in the category of profinite sets and open continuous maps, defined by the functor sending a finite set of variables into the Stone space of complete types in those variables. A model of a complete first-order theory is then a morphism from a representable simplicial set satisfying certain lifting properties reminiscent of, but weaker then, those in the definition of a fibration. The class of simplicial profinite sets corresponding to complete first order theories is characterised by the same lifting properties required of the map from the simplicial covering space (decalage) forgetting the extra degeneracy.

Paper Structure

This paper contains 46 sections, 23 equations.

Table of Contents

  1. Introduction
  2. Simplicial definitions of f.o.theories
  3. Basic notation and conventions
  4. Simplicial notations
  5. Notation for categories of spaces
  6. Vibrant simplicial maps
  7. A first order theory and its models, simplicially
  8. Introduction
  9. A first order theory in the form of Knight
  10. A $<\!\aleph_0$-saturated model of a f.o.theory in the form of Knight
  11. A f.o.theory in the form of Kamsma
  12. A model of a f.o.theory
  13. A f.o.theory as a profinite completion of its model
  14. A structure in language $\mathcal{L}$, simplicially
  15. The f.o.theory of a $<\!\aleph_0$-saturated model is its profinite compactification ?
  16. ...and 31 more sections

Theorems & Definitions (8)

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  • proof : Considerations