Morfismer

Morfismer are structure-preserving arrows between objects in a category. In category theory, a category is a collection of objects and morfismer (arrows) that go from one object to another, together with a rule for composing morfismer and with an identity morfisme for every object. The defining laws are that composition is associative and that composing any morfisme with an identity morfisme yields the same morfisme.

In concrete categories, morfismer correspond to familiar structure-preserving maps. Examples include functions between sets in the category Set, group homomorphisms between groups in Grp, ring homomorphisms between rings in Ring, continuous maps between topological spaces in Top, and linear transformations between vector spaces in Vect. Morfismer may have additional properties that reflect their context, such as preserving inverses, products, or topologies.

Key concepts related to morfismer include monomorphisms, epimorphisms and isomorphisms. A monomorfism (left-cancellable) behaves like an injective map in Set in many categories, while an epimorfism (right-cancellable) often resembles a surjective map in Set. An isomorfisme is a morfisme with an inverse morfisme, indicating that the two objects are structurally the same within the category. Not all categories have a one-to-one correspondence between monos and injective maps or between epis and surjective maps, highlighting the generality of the concept.

Morfismer form the backbone of category-theoretic reasoning. They allow the formal expression of constructions, factorization, and universal properties, and are preserved by functors, which map both objects and morfismer between categories while respecting composition and identities. Morfismer thus provide a unifying language across many areas of mathematics.