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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 ...
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 ( 1965 ) stating that if a first-order theory in a countable language is categorical in some uncountable cardinality , then it is categorical in all uncountable ...
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 ...
In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues. Categorification, when done successfully, replaces sets with categories , functions with functors , and equations with natural isomorphisms of functors satisfying additional properties.
In mathematics and statistics, a quantitative variable may be continuous or discrete if it is typically obtained by measuring or counting, respectively. [1] If it can take on two particular real values such that it can also take on all real values between them (including values that are arbitrarily or infinitesimally close together), the variable is continuous in that interval. [2]
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.
Buchholz (1986) described the following system of ordinal notation as a simplification of Feferman's theta functions. Define: Ω ξ = ω ξ if ξ > 0, Ω 0 = 1; The functions ψ v (α) for α an ordinal, v an ordinal at most ω, are defined by induction on α as follows: ψ v (α) is the smallest ordinal not in C v (α)
The monoidal product is given by concatenation of linear orders, and the unit is the empty ordinal [] (the lack of a unit prevents this from qualifying as a monoidal structure on ). In fact, Δ + {\displaystyle \Delta _{+}} is the monoidal category freely generated by a single monoid object , given by [ 0 ] {\displaystyle [0]} with the unique ...
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