Foundations of mathematics

branch of mathematics studying categories, functors, and natural transformations

study of the basic mathematical concepts

variant of ZFC, the standard axiomatic set theory

axiomatic set theory

category in mathematics where the objects are sets and the morphisms are the total functions between the sets

In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category of sets and possess a notion of localization. The Grothendieck topoi find applications in algebraic geometry, and more general elementary topoi are used in logic.

data type whose definition depends on a value

alternative foundation of mathematics

variant of type theory incorporating the univalence axiom of Voevodsky

interpretation of intuitionistic logic

relationship

generalization of category theory for higher-order morphisms

study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries