Morphismus

in the category theory (a subsection of mathematics) one regards categories so mentioned, which consist of a multiplicity of objects and Morphismen.

A category is given by two data: A class of objects and for ever two objects X and Y a classof Morphismen from X to Y. One calls Morphismen also arrows and writes f: X -> Y. To the category belongs still another partial linkage of the Morphismen, which must fulfill certain conditions, which are called in the article category theory.

In the case of one concrete category are the objects quantities with a structure and a Morphismus are a function between the which are the basis quantities, which is compatible with the structure. The linkage is in this case the usual Hintereinanderausführung of functions. In addition, there are completely differently formed categories,in which one cannot understand oneself Morphismen as functions.

examples

of examples of Morphismen are Homomorphismen of the categories, which are studied in universal algebra (e.g. Groups or rings), constant functions between topological areas, differentiable functions between differentiable variousnesses.

an identical Morphismus

• , written ID X, which is a neutral element of the composition , has _types of Morphismen each object X in each category.
• If a Morphismus f possesses a right inverse one, i.e. if it a Morphismus g with f og = ID gives, then f Retraktion is called. Similarly one names a Morphismus , which possesses a left inverse one cut (section).
• If f is both a Retraktion and a section, then f isomorphicism is called. In the case the objects X and Y can as completely homogeneous within their category to be regarded. (Isomorphicism = bijektiver Homomorphismus)
• a Morphismus from X to X is called Endomorphismus of X.
• A Endomorphismus, which is at the same time an isomorphicism, is called automorphism.
• A Morphismus f: X -> Y with the following characteristic is called Epimorphismus:

  G are, h: Y -> Z arbitrary Morphismen with g o f = h o f, then is always g = h. (e.g. each surjektive Homomorphismus is a Epimorphismus)

• a Morphismus f: X -> Y with the following characteristic is called Monomorphismus:

  G are, h: W -> X arbitrary Morphismen with f o g = f o h, then is always g = h. (e.g. if each injektive Homomorphismus is a Monomorphismus)

• is f both a Epimorphismus and a Monomorphismus, thenf is a Bimorphismus. Consider that not each Bimorphismus an isomorphicism is. It is however each Morphismus an isomorphicism, which is Epimorphismus and section, or Monomorphismus and Retraktion.

  An example of a Bimorphismus, which is not isomorphicism, supplies the imbedding of the whole numbers tothe rational numbers as Homomorphismus of rings.

• A Homöomorphismus is an isomorphicism between topological areas.
• A Diffeomorphismus is an isomorphicism between differentiable variousnesses.