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Examples include Set and CPO, the category of complete partial orders with Scott-continuous functions. A topos is a certain type of cartesian closed category in which all of mathematics can be formulated (just like classically all of mathematics is formulated in the category of sets). A topos can also be used to represent a logical theory.
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 ...
Examples include quotient spaces, direct products, completion, and duality. Many areas of computer science also rely on category theory, such as functional programming and semantics . A category is formed by two sorts of objects : the objects of the category, and the morphisms , which relate two objects called the source and the target of the ...
Abstract geometry, including algebraic geometry, categorical noncommutative geometry, etc. Quantization related to category theory, in particular categorical quantization; Categorical physics relevant for mathematics. In this article, and in category theory in general, ∞ = ω.
This is a glossary of properties and concepts in category theory in mathematics. (see also Outline of category theory.). Notes on foundations: In many expositions (e.g., Vistoli), the set-theoretic issues are ignored; this means, for instance, that one does not distinguish between small and large categories and that one can arbitrarily form a localization of a category. [1]
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics.
The categorical dual definition is a formal definition of a comonad (or cotriple); this can be said quickly in the terms that a comonad for a category is a monad for the opposite category. It is therefore a functor U {\displaystyle U} from C {\displaystyle C} to itself, with a set of axioms for counit and comultiplication that come from ...
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. The term was coined by Louis Crane. [1] [2]
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