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Several definitions and theorems about monoids may be generalized for categories. Similarly any group can be seen as a category with a single object in which every morphism is invertible , that is, for every morphism f there is a morphism g that is both left and right inverse to f under composition.
Category theory was originally introduced for the need of homological algebra, and widely extended for the need of modern algebraic geometry (scheme theory). Category theory may be viewed as an extension of universal algebra , as the latter studies algebraic structures , and the former applies to any kind of mathematical structure and studies ...
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 ...
Definition of the descent strict ω-category of a cosimplicial strict ω-category. 1991: Ross Street: Top down excision of extremals algorithm for computing nonabelian n-cocycle conditions for nonabelian cohomology. 1992: Yves Diers: Axiomatic categorical geometry using algebraic-geometric categories and algebraic-geometric functors. 1992
The language of mathematics has a wide vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject.
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
In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength.If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory.
This article provides insufficient context for those unfamiliar with the subject. Please help improve the article by providing more context for the reader, especially: For most readers, diagrams are graphical representations such as those presented in commutative diagram; the article must starts with this, and explain why the content of the article is a formalization of this representation.
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