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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 ...
Ronald Brown "Topology and Groupoids" pdf available Gives an account of some categorical methods in topology, use the fundamental groupoid on a set of base points to give a generalisation of the Seifert-van Kampen Theorem. Philip J. Higgins, "Categories and Groupoids" free download Explains some uses of groupoids in group theory and topology.
There are other modern formulations of the definition of ordinal. For example, assuming the axiom of regularity, the following are equivalent for a set x: x is a (von Neumann) ordinal, x is a transitive set, and set membership is trichotomous on x, x is a transitive set totally ordered by set inclusion, x is a transitive set of transitive sets.
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
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 ...
A geometry: it is equipped with a metric and is flat. A topology: there is a notion of open sets. There are interfaces among these: Its order and, independently, its metric structure induce its topology. Its order and algebraic structure make it into an ordered field.
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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