%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 (\(f\circ g (D_g) = f(g(D_g))\)):
Monomorphism: \(f\circ g=f\circ h \implies g=h\) (left cancellation of \(f\)) or \(f\) injective for set objects (proof)
Epimorphism: \(g\circ f=h \circ f \implies g=h\) (right cancellation) or \(f\) surjective for set objects (proof)
Isomorphism: \(f\) has \(g\) such that \(f\circ g=id_{D_g}\) and \(g \circ f = id_{D_f}\) (left inverse = right inverse) or \(f\) bijektive for set objects
Endomorphism: \(X\rightarrow X\)
Automorphism: \(X\rightarrow X\) + isomorphism
Homomorphism (Algebra): \(f(a+b)=f(a)+f(b)\) (different \(+\) possible)
Homeomorphism (Topology): \(f\) and \(f^{-1}\) continuous
Diffeomorphism (Differential geometry): bijektive, \(f\) and \(f^{-1}\) continuously differentiable