%path = "maths/morphisms" %kind = chindnum["texts"] %level = 10

The concept of a function from set theory that maps uniquely elements of one set (domain) to elements of another set (codomain), is tweaked/generalized with the concept of morphism in category theory in the sense that it puts the whole mapping in the center and combines all objects whether domain or codomain into a set of objects O. Domain and codomain in the set of objects are determined or part of a morphism (\(D_f\) is domain of f, \(C_f\) is codomain of f, both do not need to be sets). More morphisms in the set of morphisms M can share the same pair (domain, codomain). (O,M,id) is a category. id is the identity morphism.

An important aspect of a morphism is that it maintains the structure in the objects (order structure, algebraic structure, topological structure) and depending on the structure the morphisms have special names (\(f\circ g (D_g) = f(g(D_g))\)):