Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-27T20:09:23.123Z Has data issue: false hasContentIssue false
  • English
  • Français

HOMOTOPY MODEL THEORY

Part of: Applied homological algebra and category theory Model theory

Published online by Cambridge University Press:  05 October 2020

BRICE HALIMI
BRICE HALIMI*
Affiliation:
DÉPARTEMENT D’HISTOIRE ET DE PHILOSOPHIE DES SCIENCES & SPHERE UNIVERSITÉ DE PARISPARIS, FRANCEE-mail: brice.halimi@u-paris.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Drawing on the analogy between any unary first-order quantifier and a “face operator,” this paper establishes several connections between model theory and homotopy theory. The concept of simplicial set is brought into play to describe the formulae of any first-order language L, the definable subsets of any L-structure, as well as the type spaces of any theory expressed in L. An adjunction result is then proved between the category of o-minimal structures and a subcategory of the category of linearly ordered simplicial sets with distinguished vertices.

Keywords

quantifiers as face operatorsdefinable subsetstype spaceso-minimalitysimplicial setsfibrationshomotopical algebra

MSC classification

Primary: 03C64: Model theory of ordered structures; o-minimality
Secondary: 03C95: Abstract model theory 55U10: Simplicial sets and complexes
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 4 , December 2021 , pp. 1301 - 1323
Check for updates
Copyright
© Association for Symbolic Logic 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baisalov, Y. and Poizat, B., Paires de structures o-minimales, this Journal, vol. 63 (1998), no. 2, pp. 570–578.Google Scholar
Conant, J. and Thistlethwaite, O., Boolean formulae, hypergraphs and combinatorial topology. Topology and Its Applications, vol. 157 (2010), pp. 2449–2461.10.1016/j.topol.2010.08.016CrossRefGoogle Scholar
Goerss, P. G. and Jardine, J. F., Simplicial Homotopy Theory, Birkhäuser Verlag, Basel, Switzerland, 1999.10.1007/978-3-0348-8707-6CrossRefGoogle Scholar
Goodrick, J., Byunghan, K., and Kolesnikov, A., Homology groups of types in model theory and the computation of ${H}_2(p)$ , this Journal, vol. 78 (2013), no. 4, pp. 1086–1114.Google Scholar
Hofmann, M. and Streicher, T., The groupoid interpretation of type theory, Twenty Five Years of Constructive Type Theory (Sambin, G. and Smith, J. M., editors), Oxford University Press, New York, NY, 1998, pp. 83–111.Google Scholar
Hovey, M., Model Categories, American Mathematical Society, Providence, 1999.Google Scholar
Knight, R. W., Categories of topological spaces and scattered theories. Notre Dame Journal of Formal Logic, vol. 48 (2007), no. 1, pp. 53–77.10.1305/ndjfl/1172787545CrossRefGoogle Scholar
Morley, M., Categoricity in power. Transactions of the American Mathematical Society, vol. 114 (1965), no. 2, pp. 514–538.10.1090/S0002-9947-1965-0175782-0CrossRefGoogle Scholar
Pillay, A. and Steinhorn, C., Definable sets in ordered structures I. Transactions of the American Mathematical Society, vol. 295 (1986), no. 2, pp. 565–592.CrossRefGoogle Scholar
Poizat, B., A Course in Model Theory: An Introduction to Contemporary Mathematical Logic, Springer, New York, NY, 2000.10.1007/978-1-4419-8622-1CrossRefGoogle Scholar
van den Dries, L., Tame Topology and O-minimal Structures, Cambridge University Press, Cambridge, MA, 1998.10.1017/CBO9780511525919CrossRefGoogle Scholar
Weibel, C., An Introduction to Homological Algebra, Cambridge University Press, Cambridge, MA, 1995.Google Scholar