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There are many equivalent definitions of a category. [1] One commonly used definition is as follows. A category C consists of a class ob(C) of objects, a class mor(C) of morphisms or arrows, a domain or source class function dom: mor(C) → ob(C), a codomain or target class function cod: mor(C) → ob(C),
The following table classifies the various simple data types, associated distributions, permissible operations, etc. Regardless of the logical possible values, all of these data types are generally coded using real numbers, because the theory of random variables often explicitly assumes that they hold real numbers.
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 ...
Ordinal data is a categorical, statistical data type where the variables have natural, ordered categories and the distances between the categories are not known. [ 1 ] : 2 These data exist on an ordinal scale , one of four levels of measurement described by S. S. Stevens in 1946.
Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties. The term was coined by Louis Crane. [1] [2] The reverse of categorification is the process of decategorification.
This is a list of statistical procedures which can be used for the analysis of categorical data, also known as data on the nominal scale and as categorical variables. General tests [ edit ]
Each category is distinguished by properties that all its objects have in common, such as the empty set or the product of two topologies, yet in the definition of a category, objects are considered atomic, i.e., we do not know whether an object A is a set, a topology, or any other abstract concept. Hence, the challenge is to define special ...
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.
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