nLab categoricity

Idea

In model theory, given a cardinal $\kappa$, a theory is $\kappa$-categorical (or categorical in cardinality $\kappa$), if it has precisely one isomorphism class of models of cardinality $\kappa$.

Properties

The Morley categoricity theorem says that a first-order theory $T$ with countably many symbols is $\kappa$-categorical for one uncountable cardinal $\kappa$ iff $T$ is categorical in any uncountable cardinality.

The main interest is beyond first order. There is much work there, for non-elementary classes, centered around (various versions of) Shelah’s Categoricity Conjecture especially in the setting of abstract elementary classes. Also, very interesting program is initiated by Zilber in early 2000s.

Related concepts

Study of categoricity lead historically to the development of the stability theory in model theory, see also geometric stability theory.

References

• John Baldwin, Categoricity, Amer. Math. Soc. 2011, pdf

• John T. Baldwin, What is a complete theory, talk, pdf

Morley’s categoricity theorem is from

• M. Morley, Categoricity in power, Trans. Amer. Math. Soc. 114:514–538, 1965.

• wikipedia: Morley categoricity theorem

• Saharon Shelah, Classification theory and the number of non-isomorphic models, 2nd ed. North Holland Amsterdam, 1990.

• Saharon Shelah, Categoricity of abstract classes with amalgamation, Annals of Pure and Applied Logic 98(1-3) 241–294, 1999.

• Boris Zilber, Uncountably categorical theories, Transl. Math. Monog., Amer. Math. Soc. 1993

• Rami Grossberg, Monica Vandieren, Shelah’s categoricity conjecture from a successor for tame AECs, pdf

• Michael Makkai, Saharon Shelah, Categoricity of theories $L_{\kappa\omega}$ with $\kappa$ a compact cardinal, Annals of Pure and Applied Logic 47, 41-97, 1990

• Christian Espíndola, An extension of Shelah’s eventual categoricity conjecture to accessible categories with directed colimits, arxiv/1906.09169; A short proof of Shelah’s eventual categoricity conjecture for AEC’s with interpolation, under GCH, arxiv/1909.13713; A topos-theoretic proof of Shelah’s eventual categoricity conjecture, Brno, April 30, 2020, video yt, slides pdf

• Adam Harris, Categoricity and covering spaces, PhD thesis, Merton college arXiv:1412.3484

• C. Daw, A. Harris, Categoricity of modular and Shimura curves, J. Inst. Math. Jussieu, 16(5) (2017) 1075–1101 doi

• Boris Zilber, Chris Daw, Modular curves and their pseudo-analytic cover, arXiv:2107.11110; Canonical models of modular curves and the Galois action on CM-points, arXiv:2106.06387

category: model theory