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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.
a monomorphism (or monic) if f ∘ g 1 = f ∘ g 2 implies g 1 = g 2 for all morphisms g 1, g 2 : x → a. an epimorphism (or epic) if g 1 ∘ f = g 2 ∘ f implies g 1 = g 2 for all morphisms g 1, g 2 : b → x. a bimorphism if f is both epic and monic. an isomorphism if there exists a morphism g : b → a such that f ∘ g = 1 b and g ∘ f ...
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 ...
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]
In a higher topos not only mathematics can be done but also "n-geometry", which is higher homotopy theory. The topos hypothesis is that the ( n +1)-category n Cat is a Grothendieck ( n +1)-topos. Higher topos theory can also be used in a purely algebro-geometric way to solve various moduli problems in this setting.
An infinite ordinal is a regular ordinal if it is a limit ordinal that is not the limit of a set of smaller ordinals that as a set has order type less than . A regular ordinal is always an initial ordinal , though some initial ordinals are not regular, e.g., ω ω {\displaystyle \omega _{\omega }} (see the example below).
If J = 1, the category with a single object and morphism, then a diagram of shape J is essentially just an object X of C. A cone to an object X is just a morphism with codomain X . A morphism f : Y → X is a limit of the diagram X if and only if f is an isomorphism .
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.
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