functor
Quick Summary
In mathematics, a functor is a mapping between categories. Categories consist of objects and arrows (also called morphisms) that represent relationships between these objects. Functors preserve these relationships.
Think of categories as different mathematical worlds, like the world of groups or the world of sets. A functor acts like a translator, allowing you to move between these worlds while maintaining the essential structure. For example, there's a functor that maps groups to sets by simply considering the elements of the group without their operation.
This concept is a key part of category theory, a field that studies abstract structures and their relationships. Functors are fundamental for understanding how different mathematical structures are connected and how they can be generalized.1
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foundations of mathematicsAbstraction in mathematics…category of (small) categories and functors, as the morphisms between categories are called, which preserve relationships among the objects and arrows.
Read More…category of (small) categories and functors, as the morphisms between categories are called, which preserve relationships among the objects and arrows.
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algebraCategory theory…was their formulation of “functors,” a generalization of the idea of function that enabled them to connect different categories. For example, in algebraic topology functors associated topological spaces with certain groups such that their topological properties could be expressed as algebraic properties of the groups—a process that enabled powerful…
Read More…was their formulation of “functors,” a generalization of the idea of function that enabled them to connect different categories. For example, in algebraic topology functors associated topological spaces with certain groups such that their topological properties could be expressed as algebraic properties of the groups—a process that enabled powerful…
Read More