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Mathematical logic and foundations Definition

Categorical theory

Builds on TheoryModelCardinal numberCardinalityModel theory of algebraically closed fieldsSteinitz's theorem on algebraically closed fields

Definition

Let TT be a theory in a first-order signature LL, and let κ\kappa be an infinite cardinal. The theory TT is κ\kappa-categorical if two conditions hold.

  1. TT has a model whose domain has cardinality κ\kappa.
  2. Any two models of TT of cardinality κ\kappa are isomorphic.

Remarks

A theory is uncountably categorical if it is κ\kappa-categorical for every uncountable κ\kappa. Some authors drop the existence of a model of size κ\kappa from the definition.

Example: the theory ACF0\mathrm{ACF}_0 of algebraically closed fields of characteristic 00 is κ\kappa-categorical for every uncountable κ\kappa. By Steinitz's theorem, such a field is determined up to isomorphism by its transcendence degree over ℚ\Q, and a field of uncountable size κ\kappa has transcendence degree κ\kappa.

Non-example: ACF0\mathrm{ACF}_0 is not ℵ0\aleph_0-categorical. The fields ℚ‾\overline{\Q} and the algebraic closure of ℚ(t)\Q(t) are both countable models, of transcendence degree 00 and 11, so they are not isomorphic.

The theory of dense linear orders without endpoints is ℵ0\aleph_0-categorical, by Cantor's back-and-forth argument, but not κ\kappa-categorical for any uncountable κ\kappa.

Reading path

To understand this page from the ground up, read these 85 pages in order. The foundations come first.

  1. Cardinal number concept
  2. Cardinality concept
  3. Characteristic of a field concept
  4. Element definition
  5. Group concept
  6. Model concept
  7. Model theory of algebraically closed fields concept
  8. Proposition concept
  9. Theory concept
  10. Transcendence basis concept
  11. Transcendence degree concept
  12. Set definition
  13. Ring definition
  14. Implication definition
  15. Predicate definition
  16. Ordered pair definition
  17. Commutative ring definition
  18. Polynomial ring definition
  19. Ring homomorphism definition
  20. Subring definition
  21. Unit of a ring definition
  22. Zero divisor definition
  23. Ideal definition
  24. Existential quantifier definition
  25. Universal quantifier definition
  26. Cartesian product definition
  27. Divisibility in a commutative ring definition
  28. Degree of a polynomial definition
  29. Polynomial evaluation definition
  30. Field definition
  31. Kernel of a ring homomorphism definition
  32. Ring isomorphism definition
  33. Integral domain definition
  34. Subset definition
  35. Ideal generated by a subset definition
  36. Maximal ideal definition
  37. Two-sided ideal definition
  38. Uniqueness quantifier definition
  39. Leading coefficient definition
  40. Evaluation homomorphism definition
  41. Root of a polynomial definition
  42. Binary relation definition
  43. Subfield definition
  44. The only ideals of a field are zero and the whole field statement
  45. Irreducible element definition
  46. Quotient ring definition
  47. Euclidean domain definition
  48. Principal ideal definition
  49. Monic polynomial definition
  50. Split polynomial definition
  51. Degree of a product statement
  52. Field extension definition
  53. Irreducible polynomial definition
  54. Universal property of the quotient ring statement
  55. Correspondence theorem for rings statement
  56. Function definition
  57. Principal ideal domain definition
  58. Division algorithm for polynomials statement
  59. Algebraic element definition
  60. Algebraically closed field definition
  61. Intermediate field definition
  62. R[x] is an integral domain iff R is statement
  63. First isomorphism theorem for rings statement
  64. R/M is a field iff M is maximal statement
  65. Codomain definition
  66. Injective function definition
  67. In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
  68. Minimal polynomial of an algebraic element definition
  69. Algebraic extension definition
  70. Field generated by adjoining elements definition
  71. K[x] is a Euclidean domain statement
  72. Surjective function definition
  73. The minimal polynomial divides every polynomial vanishing at the element statement
  74. Algebraic closure definition
  75. Simple extension definition
  76. Splitting field definition
  77. K[x]/(f) is a field iff f is irreducible statement
  78. Bijective function definition
  79. K(a) is isomorphic to K[x] modulo the minimal polynomial statement
  80. Equinumerous sets definition
  81. Cardinality definition
  82. Uniqueness of splitting fields statement
  83. Uniqueness of the algebraic closure statement
  84. Steinitz's theorem on algebraically closed fields statement
  85. Categorical theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Categorical_theory
About this page
TagQTC57AJEVR
URIm2://QTC57AJEVR
Kinddefinition
WritingW0 stub
MSC03C35
DefinesCategorical theory
UsesTheory, Model, Cardinal number, Cardinality, Model theory of algebraically closed fields, Steinitz's theorem on algebraically closed fields
Changed2 November 2026

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