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A theory is κ-categorical (or categorical in κ) if it has exactly one model of cardinality κ up to isomorphism. Morley's categoricity theorem is a theorem of Michael D. Morley stating that if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities.
For this reason, it is used throughout mathematics. Applications to mathematical logic and semantics (categorical abstract machine) came later. Certain categories called topoi (singular topos) can even serve as an alternative to axiomatic set theory as a foundation of mathematics. A topos can also be considered as a specific type of category ...
Category theory is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas ...
There are other modern formulations of the definition of ordinal. For example, assuming the axiom of regularity, the following are equivalent for a set x: x is a (von Neumann) ordinal, x is a transitive set, and set membership is trichotomous on x, x is a transitive set totally ordered by set inclusion, x is a transitive set of transitive sets.
As a rule of thumb, an equivalence of categories preserves all "categorical" concepts and properties. If F : C → D is an equivalence, then the following statements are all true: the object c of C is an initial object (or terminal object, or zero object), if and only if Fc is an initial object (or terminal object, or zero object) of D
Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also notable for its connections to theoretical computer science. [1] In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor.
Categorical data analysis; Categorical distribution, a probability distribution; Categorical logic, a branch of category theory within mathematics with notable connections to theoretical computer science; Categorical syllogism, a kind of logical argument; Categorical proposition, a part of deductive reasoning; Categorization; Categorical perception
There is some overlap between this algebraic notion and the categorical notion of kernel since both generalize the situation of groups and modules mentioned above. In general, however, the universal-algebraic notion of kernel is more like the category-theoretic concept of kernel pair. In particular, kernel pairs can be used to interpret kernels ...